1851:
15769:
18108:: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
8801:
3572:
8576:
3912:
9144:
8311:
4002:, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."
3618:
in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in
English in a limited sense (for example, spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word
13925:
A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and
18128:
or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is
14969:
and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of
Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the
8999:
3173:
18103:
whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the
6032:
19290:
11502:
6604:
3640:
2238:
8796:{\displaystyle R/{\textstyle \bigcap _{i=1}^{n}{{\mathfrak {a}}_{i}}}\simeq \prod _{i=1}^{n}{R/{\mathfrak {a}}_{i}},\qquad x{\bmod {\textstyle \bigcap _{i=1}^{n}{\mathfrak {a}}_{i}}}\mapsto (x{\bmod {\mathfrak {a}}}_{1},\ldots ,x{\bmod {\mathfrak {a}}}_{n}).}
11203:
7822:
4010:
rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.
8043:
13839:
9571:
1812:
The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the
18496:
18713:
with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context.
15389:. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a
10825:
as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is
7262:
19208:
3530:
17954:
3463:
3399:
3335:
7918:
in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
3274:
17701:
10995:
11921:
7891:
Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the
1728:
In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a
3986:
in the definition of "ring", especially in advanced books by notable authors such as Artin, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2022 that use the term without the requirement for a
9242:
3623:). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if
15209:
13660:
2677:
2621:
16923:
4855:
3024:
17399:(by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
16718:
15417:
2479:
8497:
2423:
16157:
10814:
9146:
as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to
16413:
10148:
19123:
19042:
13396:
8557:
11337:
3050:
2010:
16293:-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a
13920:
13755:
15349:
11057:
5910:
19213:
11386:
10537:
16479:
15527:
2521:
19002:
18955:
18366:
8048:
7670:
7123:
6450:
6445:
3907:{\displaystyle {\begin{aligned}a^{3}&=4a-1,\\a^{4}&=4a^{2}-a,\\a^{5}&=-a^{2}+16a-4,\\a^{6}&=16a^{2}-8a+1,\\a^{7}&=-8a^{2}+65a-16,\\\vdots \ &\qquad \vdots \end{aligned}}}
3645:
17320:
2960:
19351:
16620:
13090:
17797:
12017:
9139:{\displaystyle R={\mathfrak {a}}_{1}\oplus \cdots \oplus {\mathfrak {a}}_{n},\quad {\mathfrak {a}}_{i}{\mathfrak {a}}_{j}=0,i\neq j,\quad {\mathfrak {a}}_{i}^{2}\subseteq {\mathfrak {a}}_{i}}
2146:
14158:
2282:
3619:
for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an
15658:
7526:
6930:
251:
19083:
13328:
9357:
5006:
2800:
12811:
10369:
9877:
6714:
4331:
2907:
2717:
2558:
2319:
15155:
14746:
13533:
7446:
3219:
543:
15716:
10411:
7404:
6781:
4751:
be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as
4020:
to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit", or "ring with 1".
1817:", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products:
11097:
12352:
7665:
18008:
14566:
6882:
12943:
9276:
8306:{\displaystyle {\begin{aligned}&(r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2})\\&(r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}\cdot r_{2},s_{1}\cdot s_{2})\end{aligned}}}
4937:
19448:
13223:
10438:
9691:
8990:
8959:
8928:
8897:
8848:
18859:
14925:
14836:
14341:
13187:
13158:
13044:
12972:
12320:
8436:
17510:
16664:
14869:
14235:
13866:
12893:
11667:
4687:
4368:
2839:
496:
459:
13122:
11837:
10575:
2870:
1145:
Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the
16823:
16766:
16232:
15573:
15277:
15084:
14377:
9970:
5594:
5565:
5532:
4732:
4283:
3998:
Gardner and
Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a
2746:
1895:
18251:
17731:
17445:
14796:
14770:
14495:
14469:
14445:
14407:
14286:
12113:
10313:
9809:
9413:
6675:
5503:
5477:
5435:
4649:
4254:
4192:
205:
12540:
3606:'s notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.
17211:
17094:
18361:
14653:
18754:
is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a
14602:
6239:
1838:
There are a few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative. For these authors, every
14197:
13448:
13016:
12644:
12592:
7118:
6323:
6268:
16543:
14058:
14013:
13701:
12750:
12685:
11378:
10688:
10046:
9729:
9645:
6294:
17164:
14965:, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the
12060:
15887:
Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.
15776:
In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a
10739:
4149:
19671:
Poonen claims that "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a
19132:
12493:
6197:
For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal
5208:
9161:
13767:
3995:
does not require unit elements in rings. In a research article, the authors often specify which definition of ring they use in the beginning of that article.
3468:
17861:
16048:
Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)
4014:
Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention:
3404:
3340:
3279:
3224:
4758:
17599:
10926:
10694:. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf.
11852:
4023:
to omit a requirement for a multiplicative identity: "rng" or "pseudo-ring", although the latter may be confusing because it also has other meanings.
16037:
3963:
gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper
15876:. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every
18030:. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the
1064:
13548:
6178:
Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite
648:
15183:
9309:(orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let
18010:
of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the
2626:
15459:
2573:
18758:. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.
18718:
between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of
17232:
16852:
16565:, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.
2965:
18193:. In particular, the algebraic geometry of the StanleyβReisner ring was used to characterize the numbers of faces in each dimension of
16623:
16673:
2428:
8448:
2375:
10747:
16349:
16088:
3168:{\displaystyle \operatorname {M} _{2}(F)=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|\ a,b,c,d\in F\right\}.}
10283:
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism
10054:
7621:. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all
1272:
equipped with two binary operations + (addition) and β
(multiplication) satisfying the following three sets of axioms, called the
19088:
19007:
13346:
8502:
11273:
18150:
12066:
6027:{\displaystyle r_{1}x_{1}+\cdots +r_{n}x_{n}\quad {\textrm {such}}\;{\textrm {that}}\;r_{i}\in R\;{\textrm {and}}\;x_{i}\in I,}
4539:
19285:{\displaystyle C^{\operatorname {op} }\to \mathbf {Rings} {\stackrel {\textrm {forgetful}}{\longrightarrow }}\mathbf {Sets} .}
11497:{\displaystyle \operatorname {End} _{R}(U)\simeq \prod _{i=1}^{r}\operatorname {M} _{m_{i}}(\operatorname {End} _{R}(U_{i})).}
1908:
21225:
21116:
21008:
20990:
20918:
20867:
20799:
20672:
20520:
20419:
20400:
3602:
defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by
20309:
15286:
11000:
6599:{\displaystyle {\begin{aligned}&f(a+b)=f(a)\ddagger f(b)\\&f(a\cdot b)=f(a)*f(b)\\&f(1_{R})=1_{S}\end{aligned}}}
4707:
as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers
2015:
The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.
10487:
5507:
contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers
105:
16418:
2486:
20594:. Carus Mathematical Monographs. Vol. 15. With an afterword by Lance W. Small. Mathematical Association of America.
18960:
18913:
2233:{\displaystyle \mathbb {Z} /4\mathbb {Z} =\left\{{\overline {0}},{\overline {1}},{\overline {2}},{\overline {3}}\right\}}
18686:, is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian
18189:. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in
15024:
is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.
2914:
1105:. (Some authors define rings without requiring a multiplicative identity and instead call the structure defined above a
19328:
16591:
13056:
1057:
17738:
11936:
21532:
21493:
21373:
21343:
21323:
21253:
21194:
21055:
20944:
20895:
20833:
20773:
20751:
20729:
20700:
20618:
20599:
20558:
19467:, such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a
15891:
14097:
2247:
641:
593:
19876:
15890:
The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the
18228:
16581:
15596:
10631:(theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in
7478:
5188:
19768:
6905:
215:
21616:
20904:
19485:
19047:
14254:
13249:
9312:
4950:
2767:
12755:
11198:{\displaystyle \operatorname {End} _{R}(\oplus _{1}^{n}U)\to \operatorname {M} _{n}(S),\quad f\mapsto (f_{ij}).}
10326:
9825:
6684:
6175:
if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.
4302:
2877:
2687:
2528:
2289:
21584:
21550:
20577:
20282:
17528:
15881:
15095:
14658:
13454:
7817:{\displaystyle {\begin{aligned}&a(x+y)=ax+ay\\&(a+b)x=ax+bx\\&1x=x\\&(ab)x=a(bx)\end{aligned}}}
7409:
4607:
3185:
1195:
510:
20078:
15663:
13871:
13706:
10374:
7362:
6732:
4610:(depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most
21638:
21555:
21525:
21108:
20791:
18031:
16721:
6976:
1050:
18129:
the tensor product. When the algebra is semisimple, the representation ring is just the character ring from
12325:
4865:
has the property of a multiplicative identity, but it is not a function and hence is not an element of
19948:
19915:
17969:
16562:
16182:
14502:
11508:
10628:
6859:
6191:
6081:
if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of
3982:
for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of
3951:
in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every
1138:. The simplest commutative rings are those that admit division by non-zero elements; such rings are called
634:
501:
18730:
Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.
12909:
9247:
4890:
20692:
19525:
19415:
17415:
such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of
15828:
15390:
13192:
10416:
9650:
8964:
8933:
8902:
8808:
5221:
3992:
919:
351:
31:
17:
8853:
4046:
The prototypical example is the ring of integers with the two operations of addition and multiplication.
21499:
21485:
20859:
18820:
18220:
18096:
16320:
15852:
15354:
14898:
14809:
14314:
13935:
13163:
13131:
13020:
12948:
12285:
111:
17477:
16640:
14845:
14202:
13844:
12816:
11632:
8398:
4658:
4344:
3975:
Fraenkel's axioms for a "ring" included that of a multiplicative identity, whereas
Noether's did not.
2814:
472:
435:
126:
19125:(multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object
18887:
13970:
viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course,
13103:
11717:
10553:
9566:{\displaystyle R=\left\{a_{n}t^{n}+a_{n-1}t^{n-1}+\dots +a_{1}t+a_{0}\mid n\geq 0,a_{j}\in R\right\}}
8570:
5805:
4403:
2848:
1780:. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called
21017:
Ballieu, R. (1947). "Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif".
18026:
were later defined in terms of homology groups in a way which is roughly analogous to the dual of a
16795:
16738:
16204:
15534:
15249:
13092:
This is the reason for the terminology "localization". The field of fractions of an integral domain
20429:
18212:
18186:
17804:
17124:
15999:
15816:
15064:
14357:
14082:
12444:
of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring
12020:
10899:
10702:
9940:
7893:
5684:
5574:
5545:
5512:
4712:
4263:
2726:
1875:
1150:
586:
389:
339:
18491:{\displaystyle {\begin{aligned}&(f+g)(x)=f(x)+g(x)\\&(f\cdot g)(x)=f(g(x)),\end{aligned}}}
18234:
17714:
17428:
14998:
consists of polynomials with integral coefficients in noncommuting variables that are elements of
14779:
14753:
14478:
14452:
14414:
14390:
14269:
12076:
10286:
9788:
6900:(that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map
6658:
5486:
5451:
5418:
4632:
4237:
4175:
4035:
188:
21593:
19615:
19505:
19500:
18190:
15402:
14962:
12498:
12138:
8372:
8011:
7589:
6186:. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left
1579:
1131:
1010:
398:
132:
91:
19846:
17179:
17062:
10864:
forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the
7257:{\displaystyle {\begin{aligned}&(a+I)+(b+I)=(a+b)+I,\\&(a+I)(b+I)=(ab)+I.\end{aligned}}}
4542:. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
4505:. The operations in this ring are addition and composition of endomorphisms. More generally, if
3978:
Most or all books on algebra up to around 1960 followed
Noether's convention of not requiring a
20587:
19886:
16264:
15834:
14615:
12437:
11541:
9883:
7897:
6950:
6819:
5236:
4167:
3956:
1802:
726:
682:
555:
406:
357:
138:
21220:, Interscience Tracts in Pure and Applied Mathematics, vol. 13, Interscience Publishers,
21050:, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press,
20431:
Applications algΓ©briques de la cohomologie des groupes, I, II, SΓ©minaire Henri Cartan, 1950/51
20292:
18742:
is the same as a ring, except that the existence of a multiplicative identity is not assumed.
14571:
10819:
together with multiplication and addition that mimic those for convergent series. It contains
6212:
6107:. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of
21131:
20319:
17057:
15784:
15758:
14346:
14167:
13409:
12977:
12605:
12553:
11926:
7569:
6299:
6244:
3179:
1250:
1183:
1098:
21313:
21184:
20270:
16484:
16251:
happens to be a field, then this is equivalent to the usual definition in field theory (cf.
14022:
13977:
13665:
12714:
12649:
11353:
10923:
As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring:
10634:
9998:
9696:
9591:
9366:
is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to
6273:
21643:
21350:
21235:
20928:
20809:
20710:
20541:
19304:
18710:
18699:
18262:
18064:
17813:
17800:
17129:
16481:
the similarity is an equivalence relation. The similarity classes with the multiplication
16015:
15382:
12025:
7551:
6942:
5769:
5292:
4862:
4436:
4399:
4064:
3620:
1839:
1094:
997:
989:
961:
956:
947:
904:
846:
279:
153:
20817:
19203:{\displaystyle h_{R}=\operatorname {Hom} (-,R):C^{\operatorname {op} }\to \mathbf {Sets} }
18702:). In essence, the most general form of a ring, is the endomorphism group of some abelian
10707:
4699:
with respect to addition β for instance, there is no natural number which can be added to
8:
19694:
19406:
18751:
18194:
18170:
18166:
18125:
18105:
18088:
17830:
17404:
16949:
16252:
15869:
15846:
12126:
10578:
8366:
with the above operations of addition and multiplication and the multiplicative identity
7994:
7828:
7565:
7061:
7011:
6371:. This latter formulation illustrates the idea of ideals as generalizations of elements.
5833:
5440:
5329:
5246:
4692:
4552:
4163:
4152:
4057:
4050:
3038:
1806:
1209:
The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by
1191:
1139:
1119:
1015:
1005:
856:
756:
748:
739:
674:
670:
561:
369:
320:
265:
159:
145:
73:
41:
14982:
The most general way to construct a ring is by specifying generators and relations. Let
14840:
It is again a field since the field operations extend to the completion. The subring of
13834:{\displaystyle 0\to M'_{\mathfrak {m}}\to M_{\mathfrak {m}}\to M''_{\mathfrak {m}}\to 0}
4093:
21572:
21464:
21412:
21289:
21279:
21243:
21156:
21044:
19520:
19472:
19398:
18276:β where rather than a vector space over a field, one has a "vector space over a ring".
18182:
18162:
18134:
18100:
18019:
17554:
16073:
15906:
15880:
domain (in particular finite division ring) is a field; in particular commutative (the
15868:
is a ring such that every non-zero element is a unit. A commutative division ring is a
15777:
12459:
12441:
10898:
to itself forms a ring with addition that is of function and multiplication that is of
9378:
9151:
6784:
5228:
5193:
4479:
4288:
4229:
3525:{\displaystyle BA=\left({\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}\right);}
2138:
1763:
below, many authors apply the term "ring" without requiring a multiplicative identity.
1238:
1135:
770:
574:
17949:{\displaystyle H^{*}(X,\mathbb {Z} )=\bigoplus _{i=0}^{\infty }H^{i}(X,\mathbb {Z} ),}
3458:{\displaystyle AB=\left({\begin{smallmatrix}0&0\\0&1\end{smallmatrix}}\right)}
3394:{\displaystyle B=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right),}
21528:
21509:
21489:
21468:
21416:
21369:
21339:
21319:
21300:
21249:
21221:
21190:
21148:
21112:
21051:
21004:
20986:
20940:
20914:
20891:
20863:
20842:
20829:
20795:
20769:
20747:
20725:
20696:
20686:
20668:
20614:
20595:
20573:
20554:
20516:
20415:
20396:
19552:
19547:
19495:
18865:
18296:
18224:
18206:
18146:
18076:
18072:
17846:
17471:
16585:
16198:
15925:
15918:
15420:. The theorem may be illustrated by the following application to linear algebra. Let
15216:
15212:
15166:
14961:
A complete ring has much simpler structure than a commutative ring. This owns to the
14954:
12902:
with respect to the complement of a prime ideal (or a union of prime ideals) in
11842:
11537:
10847:
10445:
9749:
8025:
7847:. This is not only a change of notation, as the last axiom of right modules (that is
6387:
6380:
6190:. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the
5155:
4489:
4295:
3595:
3330:{\displaystyle A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)}
2055:
1786:. Books on commutative algebra or algebraic geometry often adopt the convention that
1419:
1266:
690:
615:
412:
177:
118:
8805:
A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let
21602:
21564:
21505:
21456:
21404:
21392:
21213:
21140:
21077:
20813:
20682:
20496:
20484:
20472:
19490:
18715:
18130:
18043:
18023:
17558:
17542:
17463:
16021:
15840:
15804:
15456:
factors into powers of distinct irreducible polynomials (that is, prime elements):
15406:
14966:
12145:
11619:
11245:
9374:
7978:
7915:
6627:
5177:
3948:
3599:
3575:
3269:{\displaystyle \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)}
2107:
1782:
1736:
1588:
1463:
1242:
1218:
1210:
1187:
1154:
1123:
1102:
841:
694:
621:
607:
421:
363:
326:
99:
85:
21338:, Graduate Studies in Mathermatics, vol. 145, American Mathematical Society,
21205:
10695:
866:
30:
This article is about the algebraic structure. For other uses in mathematics, see
21333:
21231:
21180:
21126:
20924:
20908:
20805:
20706:
20660:
20644:
20628:
20537:
19601:
19572:
19461:
18891:
18158:
17853:
16927:
16235:
16007:
15931:
15810:
15386:
15358:
13931:
13244:
11512:
9739:
9581:
9394:
6183:
6135:
4696:
4443:, forms a ring with matrix addition and matrix multiplication as operations. For
4080:
3591:
3566:
3034:
1767:
1234:
1226:
1199:
1146:
933:
927:
914:
894:
885:
851:
788:
722:
383:
333:
171:
17696:{\displaystyle \lambda ^{n}(x+y)=\sum _{0}^{n}\lambda ^{i}(x)\lambda ^{n-i}(y).}
16189:
Semisimplicity is closely related to separability. A unital associative algebra
13938:
amounts to making some morphisms isomorphisms. An element in a commutative ring
10990:{\displaystyle \operatorname {End} _{R}(R^{n})\simeq \operatorname {M} _{n}(R).}
10856:
be a ring (not necessarily commutative). The set of all square matrices of size
9576:
forms a commutative ring with the usual addition and multiplication, containing
9373:
An important application of an infinite direct product is the construction of a
7618:
6991:, on the other hand, is not always an ideal, but it is always a subring of
6179:
3487:
3423:
3356:
3295:
3234:
20529:
20468:
19611:
18154:
17964:
17817:
17591:
17385:
17023:
16939:
16083:
15898:
15157:
then the resulting ring will be the usual polynomial ring with coefficients in
14971:
14303:
12149:
11916:{\displaystyle {\overline {\mathbf {F} }}_{p}=\varinjlim \mathbf {F} _{p^{m}}.}
7045:
6164:
form an ideal of the integers, and this ideal is generated by the integer
4625:
4611:
4007:
1863:
975:
714:
702:
427:
21408:
17357:
is the ring of integers, then we recover the previous example (by identifying
16930:
generalize the notion of central simple algebras to a commutative local ring.
15418:
structure theorem for finitely generated modules over a principal ideal domain
4744:(which has all of the axioms of a ring excluding that of an additive inverse).
4049:
The rational, real and complex numbers are commutative rings of a type called
21632:
21424:
21361:
21357:
21152:
21100:
21082:
21065:
20913:, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Springer,
20761:
20739:
20717:
20456:
19684:
Some other authors such as Lang further require a zero divisor to be nonzero.
19654:
19576:
19542:
19468:
19402:
18739:
18092:
17396:
16069:
15945:
15935:
is a ring that is semisimple as a left module (or right module) over itself.
15902:
15865:
15792:
15730:
15590:
15394:
13051:
12130:
10831:
7585:
7041:
7035:
6187:
5599:
5399:
can be equipped with operations making it a ring such that the inclusion map
4475:
4380:
3615:
1730:
1382:
1340:
1284:
1230:
1214:
1168:
1090:
861:
826:
783:
568:
464:
79:
20878:
15872:. A prominent example of a division ring that is not a field is the ring of
4521:-linear maps forms a ring, also called the endomorphism ring and denoted by
1093:
with respect to the addition operator, and the multiplication operator is
21440:
21263:
21167:
19596:
19591:
19586:
19581:
19537:
19510:
18584:
18273:
18121:
18068:
18027:
17408:
17381:
17222:
17221:
whose supports (the sets of points at which the functions are nonzero) are
16546:
16029:
15410:
15378:
14801:
11846:
10459:
to itself; the addition and the multiplication are those of functions. Let
7568:) by generalizing from multiplication of vectors with elements of a field (
7561:
6826:
5125:
5061:
As a special case, one can define nonnegative integer powers of an element
4407:
4003:
3960:
3952:
3603:
1754:
1222:
1035:
966:
800:
730:
600:
375:
271:
21607:
21588:
21522:
Rings and things and a fine array of twentieth century associative algebra
19635:
This means that each operation is defined and produces a unique result in
18038:, which is analogous to the observation that one can multiply pointwise a
15002:. A free ring satisfies the universal property: any function from the set
10178:
be a polynomial in one variable, that is, an element in a polynomial ring
7572:) to multiplication with elements of a ring. More precisely, given a ring
20746:. Graduate Texts in Mathematics. Vol. 131 (2nd ed.). Springer.
20210:
19557:
19378:
18755:
18035:
17960:
17226:
16631:
16282:
16033:
15978:
15949:
15398:
15393:(UFD), an integral domain in which every nonunit element is a product of
12152:) of rings is defined as follows. Suppose we are given a family of rings
10865:
10843:
9237:{\displaystyle 1=e_{1}+\cdots +e_{n},\quad e_{i}\in {\mathfrak {a}}_{i}.}
6897:
6618:
6206:
6172:
4752:
4574:
4510:
4006:
makes the counterargument that the natural notion for rings would be the
3579:
3553:
1127:
1118:
Whether a ring is commutative has profound implications on its behavior.
1025:
1020:
909:
899:
873:
678:
662:
580:
291:
165:
47:
21293:
21267:
8384:. The same construction also works for an arbitrary family of rings: if
7885:, if left multiplication (by ring elements) is used for a right module.
6194:). The integers, however, form a Noetherian ring which is not Artinian.
3970:
1157:
of a number field. Examples of noncommutative rings include the ring of
21576:
21460:
21174:. Vol. 2: Seminumerical Algorithms (3rd ed.). AddisonβWesley.
21160:
20783:
19567:
18117:
17342:
17341:. The subring consisting of elements with finite support is called the
15873:
15822:
13047:
10827:
9382:
5444:
4560:
1179:
1130:. Its development has been greatly influenced by problems and ideas of
775:
718:
345:
19658:
is used if existence of a multiplicative identity is not assumed. See
18722:
closed under addition and under composition with arbitrary morphisms.
15204:{\displaystyle S\mapsto {\text{the free ring generated by the set }}S}
13332:
The most important properties of localization are the following: when
4587:
as basis. Multiplication is defined by the rules that the elements of
21543:
21039:
20794:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
20508:
20183:
18528:, and function composition is denoted from right to left. Therefore,
18140:
18034:. However, the advantage of the cohomology groups is that there is a
15374:
14987:
6631:
5181:
4391:
4197:
2672:{\displaystyle {\overline {3}}\cdot {\overline {3}}={\overline {1}}.}
2089:
1798:
1030:
836:
793:
761:
305:
210:
21568:
21444:
21144:
21129:(1945), "Structure theory of algebraic algebras of bounded degree",
21003:, Mathematics and its Applications, Chichester: Ellis Horwood Ltd.,
20985:, Mathematics and its Applications, Chichester: Ellis Horwood Ltd.,
17565:
7330:. It is surjective and satisfies the following universal property:
6134:
are left ideals and right ideals, respectively; they are called the
2616:{\displaystyle {\overline {2}}\cdot {\overline {3}}={\overline {2}}}
2023:
Some basic properties of a ring follow immediately from the axioms:
1245:. They later proved useful in other branches of mathematics such as
20171:
19606:
19562:
19515:
18767:
18719:
18169:
to study geometric concepts in terms of ring-theoretic properties.
17803:. The notion plays a central rule in the algebraic approach to the
16580:
is a finite field or an algebraically closed field (more generally
15787:
that describes the relationship between rings, domains and fields:
9748:
is also an integral domain; its field of fractions is the field of
4741:
1859:
1797:
In a ring, multiplicative inverses are not required to exist. A non
1246:
1203:
831:
698:
299:
285:
21284:
20348:
19652:
The existence of 1 is not assumed by some authors; here, the term
16918:{\displaystyle H^{2}\left(\operatorname {Gal} (F/k),k^{*}\right).}
15413:, where an "ideal" admits prime factorization, fails to be a PID.
6609:
If one is working with rngs, then the third condition is dropped.
4850:{\displaystyle (f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy.}
3019:{\displaystyle -{\overline {3}}={\overline {-3}}={\overline {1}}.}
1757:
with the usual + and β
is a rng, but not a ring. As explained in
21395:(1915). "Γber die Teiler der Null und die Zerlegung von Ringen".
20258:
20222:
18529:
17834:
13959:
7888:
Basic examples of modules are ideals, including the ring itself.
6156:. For example, the set of all positive and negative multiples of
5318:
5304:
1855:
710:
706:
183:
67:
20246:
20234:
19903:
16713:{\displaystyle \operatorname {Br} (k)=\mathbb {Q} /\mathbb {Z} }
15768:
10481:. The universal property says that this map extends uniquely to
2474:{\displaystyle {\overline {3}}+{\overline {3}}={\overline {2}}.}
19710:
18011:
15416:
Among theorems concerning a PID, the most important one is the
8492:{\displaystyle {\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}}
6423:
that preserves the ring operations; namely, such that, for all
2721:
is a ring: each axiom follows from the corresponding axiom for
2418:{\displaystyle {\overline {2}}+{\overline {3}}={\overline {1}}}
1900:
1870:
The most familiar example of a ring is the set of all integers
1479:
765:
16323:
states any automorphism of a central simple algebra is inner.
16152:{\textstyle \prod _{i=1}^{r}\operatorname {M} _{n_{i}}(D_{i})}
10809:{\displaystyle \sum _{0}^{\infty }a_{i}t^{i},\quad a_{i}\in R}
10448:
satisfies the universal property and so is a polynomial ring.
1850:
20159:
19722:
18660:. It is in fact true that this association of any element of
18532:
to any abelian group, is a ring. Conversely, given any ring,
18095:
which uses a ring to describe the various ways the group can
18063:
The ring structure in cohomology provides the foundation for
18015:
16408:{\displaystyle A\otimes _{k}k_{n}\approx B\otimes _{k}k_{m}.}
14345:
The completion can in this case be constructed also from the
12898:
The localization is frequently applied to a commutative ring
10471:
defines a constant function, giving rise to the homomorphism
8770:
8739:
8685:
7997:. In particular, every ring is an algebra over the integers.
7105:
6168:. In fact, every ideal of the ring of integers is principal.
5824:; they (each individually) generate a subring of the center.
5678:
is the smallest positive integer such that this occurs, then
21368:. Graduate Texts in Mathematics. Vol. 28β29. Springer.
20858:. Cambridge Studies in Advanced Mathematics (2nd ed.).
12169:
running over positive integers, say, and ring homomorphisms
10143:{\displaystyle k\to k,\,f\mapsto f\left(t^{2},t^{3}\right).}
7298:
As with a quotient group, there is a canonical homomorphism
3571:
20089:
19966:
19891:
19118:{\displaystyle \operatorname {pt} {\stackrel {1}{\to }}\,R}
19037:{\displaystyle \operatorname {pt} {\stackrel {0}{\to }}\,R}
18099:
on a finite set. The
Burnside ring's additive group is the
14160:
it is a commutative ring. The canonical homomorphisms from
13391:{\displaystyle {\mathfrak {p}}\mapsto {\mathfrak {p}}\left}
8552:{\displaystyle {\mathfrak {a}}_{i}+{\mathfrak {a}}_{j}=(1)}
8037:
can be equipped with the following natural ring structure:
3085:
1745:
21032:
20724:. Graduate Texts in Mathematics. Vol. 189. Springer.
20536:. Graduate Texts in Mathematics. Vol. 150. Springer.
20297:
18591:, that "factor through" right (or left) multiplication of
16258:
11332:{\displaystyle U=\bigoplus _{i=1}^{r}U_{i}^{\oplus m_{i}}}
8438:
is a ring with componentwise addition and multiplication.
7922:
Any ring homomorphism induces a structure of a module: if
1742:
21589:"From Numbers to Rings: The Early History of Ring Theory"
21318:, Lecture Notes in Mathematics, vol. 585, Springer,
21248:. Graduate Texts in Mathematics. Vol. 88. Springer.
20768:. Problem Books in Mathematics (2nd ed.). Springer.
20534:
Commutative algebra with a view toward algebraic geometry
20325:
19977:
19791:
18173:
studies maps between the subrings of the function field.
17840:
17824:
17561:, and in either case one would obtain a topological ring.
17553:
matrices over the real numbers could be given either the
5605:
An intersection of subrings is a subring. Given a subset
5321:
if any one of the following equivalent conditions holds:
1770:, ring multiplication is not required to be commutative:
20036:
19938:
19936:
19803:
15593:, each of which is isomorphic to the module of the form
5812:. The center is the centralizer of the entire ring
5239:; in this case the inverse is unique, and is denoted by
2005:{\displaystyle \dots ,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots }
21307:, Graduate Texts in Mathematics, vol. 67, Springer
20572:. Chapman & Hall/CRC Pure and Applied Mathematics.
20395:(2nd ed.). Cambridge: Cambridge University Press.
20135:
19129:
equipped with a factorization of its functor of points
14990:(that is, free algebra over the integers) with the set
13655:{\displaystyle 0\to M'\left\to M\left\to M''\left\to 0}
5287:
consists of the set of all invertible matrices of size
4465:
not the zero ring), this matrix ring is noncommutative.
20200:
20198:
20147:
20024:
20012:
20000:
19471:
in a good category of spectra such as the category of
18111:
16955:
is a group homomorphism from the multiplicative group
16091:
15344:{\displaystyle (x\otimes u)(y\otimes v)=xy\otimes uv.}
14977:
13398:
is a bijection between the set of all prime ideals in
12329:
11052:{\displaystyle f:\oplus _{1}^{n}U\to \oplus _{1}^{n}U}
8856:
8689:
8589:
8401:
3947:
The first axiomatic definition of a ring was given by
3092:
1801:
commutative ring in which every nonzero element has a
1101:
over the addition operation, and has a multiplicative
20633:
20111:
20048:
19933:
19853:
19418:
19331:
19216:
19135:
19091:
19050:
19010:
18963:
18916:
18823:
18364:
18237:
17972:
17864:
17741:
17717:
17602:
17480:
17431:
17235:
17182:
17132:
17065:
16855:
16798:
16741:
16676:
16643:
16594:
16487:
16421:
16352:
16207:
15666:
15599:
15537:
15462:
15289:
15252:
15186:
15098:
15067:
14901:
14848:
14812:
14782:
14756:
14661:
14618:
14574:
14505:
14481:
14455:
14417:
14393:
14360:
14317:
14272:
14205:
14170:
14100:
14025:
13980:
13874:
13847:
13770:
13709:
13668:
13551:
13457:
13412:
13349:
13252:
13195:
13166:
13134:
13106:
13059:
13023:
12980:
12951:
12912:
12819:
12758:
12717:
12652:
12608:
12556:
12501:
12462:
12328:
12288:
12079:
12028:
11939:
11855:
11720:
11635:
11389:
11356:
11276:
11100:
11003:
10929:
10750:
10710:
10637:
10556:
10532:{\displaystyle R\to S,\quad f\mapsto {\overline {f}}}
10490:
10419:
10377:
10329:
10289:
10057:
10001:
9943:
9828:
9791:
9699:
9653:
9594:
9416:
9315:
9250:
9164:
9002:
8967:
8936:
8905:
8811:
8579:
8505:
8451:
8046:
7668:
7481:
7412:
7365:
7121:
6908:
6862:
6735:
6687:
6661:
6448:
6302:
6276:
6247:
6215:
5913:
5577:
5548:
5515:
5489:
5454:
5421:
5350:
a ring with the same multiplicative identity as
5196:
5136:
in the ring such that there exists a nonzero element
4953:
4893:
4761:
4715:
4661:
4635:
4347:
4305:
4266:
4240:
4178:
4096:
3643:
3471:
3407:
3343:
3282:
3227:
3188:
3053:
2968:
2917:
2880:
2851:
2817:
2770:
2729:
2690:
2629:
2576:
2531:
2489:
2431:
2378:
2292:
2250:
2149:
1911:
1878:
1225:. Rings were first formalized as a generalization of
717:, but they may also be non-numerical objects such as
513:
475:
438:
218:
191:
21553:(1996). "The Genesis of the Abstract Ring Concept".
21429:
Jahresbericht der
Deutschen Mathematiker-Vereinigung
21427:(1897). "Die Theorie der algebraischen ZahlkΓΆrper".
20360:
19988:
19827:
19779:
19749:
19739:
19737:
18810:(since it no longer follows from the other axioms).
18176:
17545:
or any other product in the category). For example,
16474:{\displaystyle k_{n}\otimes _{k}k_{m}\simeq k_{nm},}
15772:
Hierarchy of several classes of rings with examples.
15522:{\displaystyle q=p_{1}^{e_{1}}\ldots p_{s}^{e_{s}}.}
11669:
defined as follows: it is the disjoint union of all
11548:
is equivalent to the category of right modules over
10837:
6998:
To give a ring homomorphism from a commutative ring
6099:
is a left ideal, called the left ideal generated by
5655:. It can be generated through addition of copies of
5249:
under ring multiplication; this group is denoted by
3532:
this example shows that the ring is noncommutative.
2516:{\displaystyle {\overline {x}}\cdot {\overline {y}}}
1739:
1081:is a set endowed with two binary operations called
27:
Algebraic structure with addition and multiplication
20195:
20123:
20060:
19865:
18997:{\displaystyle R\times R\;{\stackrel {m}{\to }}\,R}
18950:{\displaystyle R\times R\;{\stackrel {a}{\to }}\,R}
18790:is a commutative monoid, and adding the axiom that
18157:contained in the function field and containing the
18153:. The points of an algebraic variety correspond to
13160:is the same as the residue field of the local ring
12115:where the limit runs over all the coordinate rings
5816:. Elements or subsets of the center are said to be
21186:Mathematical Handbook for Scientists and Engineers
21043:
20649:American Mathematical Society Mathematical Surveys
20100:
19815:
19442:
19345:
19284:
19202:
19117:
19077:
19036:
18996:
18949:
18853:
18490:
18245:
18141:Function field of an irreducible algebraic variety
18002:
17948:
17791:
17725:
17695:
17504:
17439:
17315:{\displaystyle (f*g)(t)=\sum _{s\in G}f(s)g(t-s).}
17314:
17205:
17158:
17088:
16917:
16817:
16760:
16712:
16658:
16614:
16537:
16473:
16407:
16226:
16151:
16036:, but it is not semisimple. The same holds for a
15710:
15652:
15567:
15521:
15424:be a finite-dimensional vector space over a field
15343:
15271:
15203:
15149:
15078:
14919:
14863:
14830:
14790:
14764:
14740:
14647:
14596:
14560:
14489:
14463:
14439:
14401:
14371:
14335:
14280:
14229:
14191:
14152:
14052:
14007:
13914:
13860:
13833:
13749:
13695:
13654:
13527:
13442:
13390:
13322:
13217:
13181:
13152:
13116:
13084:
13038:
13010:
12966:
12937:
12887:
12805:
12744:
12679:
12638:
12586:
12534:
12487:
12346:
12314:
12107:
12054:
12011:
11915:
11831:
11661:
11496:
11372:
11331:
11197:
11051:
10989:
10808:
10733:
10682:
10569:
10531:
10432:
10405:
10363:
10307:
10142:
10040:
9964:
9871:
9803:
9723:
9685:
9639:
9565:
9351:
9270:
9236:
9138:
8984:
8953:
8922:
8891:
8842:
8795:
8551:
8491:
8430:
8305:
7816:
7520:
7440:
7398:
7256:
6924:
6876:
6775:
6708:
6669:
6598:
6317:
6288:
6262:
6233:
6026:
5588:
5559:
5526:
5497:
5471:
5443:of real numbers and also a subring of the ring of
5429:
5202:
5000:
4931:
4849:
4726:
4681:
4643:
4362:
4325:
4277:
4248:
4186:
4143:
3906:
3539:, commutative or not, and any nonnegative integer
3524:
3457:
3393:
3329:
3268:
3213:
3167:
3018:
2955:{\displaystyle -{\overline {x}}={\overline {-x}}.}
2954:
2901:
2864:
2833:
2794:
2740:
2711:
2671:
2615:
2552:
2515:
2473:
2417:
2313:
2276:
2232:
2004:
1889:
537:
490:
453:
245:
199:
21335:The K-book: An Introduction to Algebraic K-theory
21093:Handbook of Mathematics and Computational Science
20998:
20980:
20611:Abstract Algebra: an Introduction, Second Edition
20337:
19734:
19659:
19346:{\displaystyle \operatorname {Spec} \mathbb {Z} }
18200:
17780:
17767:
17225:. It is a field with the multiplication given by
17199:
17189:
17149:
17139:
17082:
17072:
16615:{\displaystyle \operatorname {Br} (\mathbb {R} )}
13085:{\displaystyle {\mathfrak {p}}R_{\mathfrak {p}}.}
12045:
12035:
12005:
11995:
11956:
11946:
10997:This is a special case of the following fact: If
10727:
10717:
10230:in that ring. The result of substituting zero to
3923:is going to be an integral linear combination of
3590:The study of rings originated from the theory of
1111:
21630:
21504:
21484:Bronshtein, I. N. and Semendyayev, K. A. (2004)
20999:Balcerzyk, StanisΕaw; JΓ³zefiak, Tadeusz (1989),
20981:Balcerzyk, StanisΕaw; JΓ³zefiak, Tadeusz (1989),
20841:
19728:
17792:{\displaystyle \lambda ^{n}(x)={\binom {x}{n}},}
16038:ring of differential operators in many variables
15223:(and it is often called the free ring functor.)
14608:in the prime factorization of a nonzero integer
13946:-module. Thus, categorically, a localization of
12012:{\displaystyle k(\!(t)\!)=\varinjlim t^{-m}k\!]}
11714:A polynomial ring in infinitely many variables:
11566:correspond in one-to-one to two-sided ideals in
10585:. The resulting map is injective if and only if
9403:(called a variable) and a commutative ring
3614:The term "Zahlring" (number ring) was coined by
21615:
21001:Dimension, multiplicity and homological methods
20903:
20567:
20467:
20276:
20095:
19972:
18774:) is obtained by weakening the assumption that
18272:-module is a generalization of the notion of a
16837:.) If the extension is finite and Galois, then
15757:and the above decomposition corresponds to the
15180:In the category-theoretic terms, the formation
14153:{\displaystyle {\hat {R}}=\varprojlim R/I^{n};}
11583:
9377:of rings (see below). Another application is a
9158:has the above decomposition. Then we can write
5540:and thus does not qualify as a subring of
4653:with the usual operations is not a ring, since
4398:becomes a ring if we define addition to be the
3276:is the multiplicative identity of the ring. If
2277:{\displaystyle {\overline {x}}+{\overline {y}}}
2030:The additive inverse of each element is unique.
1753:) with a missing "i". For example, the set of
1723:
21500:History of ring theory at the MacTutor Archive
21029:
15283:-algebra with multiplication characterized by
15027:Now, we can impose relations among symbols in
14249:is a Noetherian local ring with maximal ideal
10701:There are some other related constructions. A
9150:. Equivalently, the above can be done through
6798:be a commutative ring of prime characteristic
4597:and multiply together as they do in the group
4211:be a ring. Then the set of all functions from
21356:
21090:
21046:Skew Fields: Theory of General Division Rings
20962:
19983:
18335:, the following rules may be used to compute
18185:has an associated face ring, also called its
17390:
15653:{\displaystyle k/\left(p_{i}^{k_{j}}\right).}
14257:. The construction is especially useful when
12546:; that is, the homomorphism maps elements in
7521:{\displaystyle {\overline {f}}:R/\ker f\to S}
7022:(which in particular gives a structure of an
5151:. A right zero divisor is defined similarly.
4170:defined on the real line forms a commutative
3221:satisfies the above ring axioms. The element
2132:
1058:
642:
20551:Contemporary Abstract Algebra, Sixth Edition
18877:
18848:
18824:
18552:is an abelian group. Furthermore, for every
17833:can be fruitfully analyzed in terms of some
15141:
15105:
13942:may be thought of as an endomorphism of any
13542:with partial ordering given by divisibility.
13124:is a prime ideal of a commutative ring
8573:says there is a canonical ring isomorphism:
6925:{\displaystyle \mathbb {Z} \to \mathbb {Q} }
5220:. One example of an idempotent element is a
697:satisfying properties analogous to those of
246:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
21524:. Mathematical Surveys and Monographs, 65.
19078:{\displaystyle R\;{\stackrel {i}{\to }}\,R}
18782:is an abelian group to the assumption that
18082:
13323:{\displaystyle M\left=R\left\otimes _{R}M.}
9772:is a unique factorization domain. Finally,
9352:{\displaystyle {\mathfrak {a}}_{i}=Re_{i},}
6622:if there exists an inverse homomorphism to
6103:; it is the smallest left ideal containing
5277:is the ring of all square matrices of size
5001:{\displaystyle P_{n}=\prod _{i=1}^{n}a_{i}}
3971:Multiplicative identity and the term "ring"
3178:With the operations of matrix addition and
3028:
2795:{\displaystyle \mathbb {Z} /4\mathbb {Z} ,}
21063:
20956:Introduction to Foundations of Mathematics
20608:
20393:Galois Theory and Its Algebraic Background
20042:
19054:
18973:
18926:
18872:
18861:with ordinary addition and multiplication;
17531:as maps between topological spaces (where
15725:) has a basis in which the restriction of
13966:-modules to itself that sends elements of
12806:{\displaystyle r/f^{n},\,r\in R,\,n\geq 0}
12594:and, moreover, any ring homomorphism from
12422:For an example of a projective limit, see
10364:{\displaystyle {\overline {\phi }}:R\to S}
9872:{\displaystyle R\to S,\quad f\mapsto f(x)}
6983:. The kernel is a two-sided ideal of
6709:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
6138:left ideals and right ideals generated by
6004:
5996:
5979:
5971:
5176:. One example of a nilpotent element is a
4326:{\displaystyle {\widehat {\mathbb {Z} }},}
2902:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
2841:which is consistent with the notation for
2712:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
2553:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
2314:{\displaystyle \mathbb {Z} /4\mathbb {Z} }
2099:contains the zero ring as a subring, then
1866:, form the prototypical example of a ring.
1065:
1051:
649:
635:
21606:
21283:
21081:
20853:
20681:
19339:
19314:-schemes. One example is the ring scheme
19111:
19071:
19030:
18990:
18943:
18572:, by right (or left) distributivity. Let
18239:
17993:
17936:
17885:
17719:
17433:
17115:is the least degree of a nonzero term in
16706:
16696:
16646:
16605:
15585:-module. The structure theorem then says
15363:
15150:{\displaystyle E=\{xy-yx\mid x,y\in X\},}
15069:
14904:
14851:
14815:
14784:
14758:
14741:{\displaystyle |m/n|_{p}=|m|_{p}/|n|_{p}}
14483:
14457:
14395:
14362:
14320:
14274:
13528:{\displaystyle R\left=\varinjlim R\left,}
12793:
12780:
10094:
9980:; it is the same thing as the subring of
9819:, one can consider the ring homomorphism
7896:). In particular, not all modules have a
7441:{\displaystyle f={\overline {f}}\circ p.}
7010:is the same as to give a structure of an
6971:, the set of all elements mapped to 0 by
6918:
6910:
6864:
6702:
6689:
6663:
6626:(that is, a ring homomorphism that is an
5582:
5550:
5520:
5491:
5456:
5423:
4837:
4740:) form an algebraic structure known as a
4717:
4666:
4637:
4350:
4310:
4268:
4242:
4200:addition and multiplication of functions.
4180:
3214:{\displaystyle \operatorname {M} _{2}(F)}
2895:
2882:
2785:
2772:
2731:
2705:
2692:
2546:
2533:
2307:
2294:
2164:
2151:
1880:
538:{\displaystyle \mathbb {Z} (p^{\infty })}
515:
478:
441:
239:
226:
193:
21477:
21391:
21311:
21178:
21125:
20659:
20643:
20627:
20586:
20528:
20513:Basic algebra: groups, rings, and fields
20495:
20483:
20366:
20354:
20240:
20189:
20177:
20153:
20141:
20018:
20006:
19954:
19921:
19909:
19897:
19716:
17820:that is compatible with ring operations.
15767:
15711:{\displaystyle p_{i}(t)=t-\lambda _{i},}
15195:the free ring generated by the set
14931:Similarly, the formal power series ring
14237:The latter homomorphism is injective if
13915:{\displaystyle 0\to M'\to M\to M''\to 0}
13750:{\displaystyle 0\to M'\to M\to M''\to 0}
11067:may be written as a matrix with entries
10902:; it is called the endomorphism ring of
10406:{\displaystyle {\overline {\phi }}(t)=x}
10323:there exists a unique ring homomorphism
10153:In other words, it is the subalgebra of
7399:{\displaystyle {\overline {f}}:R/I\to S}
6776:{\displaystyle R\to R,x\mapsto uxu^{-1}}
5792:that commute with every element in
4383:, the ring generated by Hecke operators.
3570:
2088:has only one element, and is called the
1849:
1591:with respect to addition, meaning that:
32:Ring (disambiguation) Β§ Mathematics
21583:
21549:
21538:ItΓ΄, K. editor (1986) "Rings." Β§368 in
21439:
21423:
21030:Berrick, A. J.; Keating, M. E. (2000).
21016:
20963:Zariski, Oscar; Samuel, Pierre (1958).
20548:
20390:
20066:
20030:
19942:
19871:
19859:
19839:
18071:, intersection theory on manifolds and
17337:is the least element in the support of
16259:Central simple algebra and Brauer group
15397:(an element is prime if it generates a
14264:The basic example is the completion of
12137:Any commutative ring is the colimit of
9811:be commutative rings. Given an element
9766:is a unique factorization domain, then
7465:, invoking the universal property with
6854:, there are a unique ring homomorphism
5699:is never zero for any positive integer
5621:is the intersection of all subrings of
4414:
4335:the (infinite) product of the rings of
3942:
1700:In notation, the multiplication symbol
709:. Ring elements may be numbers such as
14:
21631:
21540:Encyclopedic Dictionary of Mathematics
21331:
21262:
21241:
21212:
21099:
20983:Commutative Noetherian and Krull rings
20953:
20934:
20885:
20744:A first course in noncommutative rings
20315:
20129:
20117:
20106:
20054:
19785:
19755:
18745:
17841:Cohomology ring of a topological space
17825:Some examples of the ubiquity of rings
15929:is a direct sum of simple modules. A
15829:unique factorization domains
13926:a sheaf is inherently a local notion.)
12347:{\displaystyle \textstyle \prod R_{i}}
12067:function field of an algebraic variety
10690:and the set of closed subvarieties of
10627:be an algebraically closed field. The
10592:Given a non-constant monic polynomial
10048:denotes the image of the homomorphism
7359:, then there is a unique homomorphism
7006:with image contained in the center of
5536:does not contain the identity element
4878:
4540:endomorphism ring of an elliptic curve
4219:forms a ring, which is commutative if
2804:and this element is often denoted by "
2033:The multiplicative identity is unique.
1237:and rings of invariants that occur in
21299:
21166:
21139:(4), Annals of Mathematics: 695β707,
20973:
20876:
20667:. Vol. 1 (2nd ed.). Dover.
20455:
20448:
20437:
20427:
20343:
20264:
20252:
19994:
19833:
19821:
19639:for each ordered pair of elements of
18299:(see above). Note that, essentially,
18003:{\displaystyle H_{i}(X,\mathbb {Z} )}
16277:-algebra is central if its center is
15442:a linear map with minimal polynomial
15088:then the resulting ring will be over
15031:by taking a quotient. Explicitly, if
14748:). It defines a distance function on
14561:{\displaystyle |n|_{p}=p^{-v_{p}(n)}}
12019:(it is the field of fractions of the
11562:. In particular, two-sided ideals in
11091:, resulting in the ring isomorphism:
9693:forms a commutative ring, containing
8961:are rings though not subrings). Then
6877:{\displaystyle \mathbb {Z} \mapsto R}
5119:
21488:, 4th ed. New York: Springer-Verlag
21203:
21038:
20823:
20782:
20568:Gardner, J.W.; Wiegandt, R. (2003).
20507:
20409:
20331:
20303:
20288:
20228:
20216:
20204:
20165:
19882:
19809:
19769:"Non-associative rings and algebras"
19743:
19693:Such a central idempotent is called
19310:is a ring object in the category of
18560:, right (or left) multiplication by
17172:and a totally ordered abelian group
15385:. A commutative domain is called an
15238:be algebras over a commutative ring
14241:is a Noetherian integral domain and
12938:{\displaystyle S=R-{\mathfrak {p}},}
12495:together with the ring homomorphism
12440:generalizes the construction of the
9359:which are two-sided ideals. If each
9271:{\displaystyle {\mathfrak {a}}_{i},}
7914:, where the first minus denotes the
6641:The function that maps each integer
4932:{\displaystyle (a_{1},\dots ,a_{n})}
4450:, this matrix ring is isomorphic to
4040:
2058:with respect to multiplication) and
1482:under multiplication, meaning that:
106:Free product of associative algebras
21516:(5th ed.), New York: Macmillan
21091:Harris, J. W.; Stocker, H. (1998).
20760:
20738:
20716:
20477:Introduction to commutative algebra
20279:, Theorem 10.17 and its corollaries
19797:
19443:{\displaystyle \mu :X\wedge X\to X}
18112:Representation ring of a group ring
17034:consisting of zero and all nonzero
16961:to a totally ordered abelian group
16622:has order 2 (a special case of the
15912:
15817:integrally closed domains
15211:is the left adjoint functor of the
15047:is called the ring with generators
14978:Rings with generators and relations
13850:
13816:
13801:
13783:
13406:and the set of all prime ideals in
13362:
13352:
13218:{\displaystyle k({\mathfrak {p}}).}
13204:
13173:
13145:
13109:
13073:
13062:
13030:
12958:
12927:
11676:'s modulo the equivalence relation
10444:. For example, choosing a basis, a
10433:{\displaystyle {\overline {\phi }}}
9686:{\displaystyle t_{1},\ldots ,t_{n}}
9319:
9254:
9220:
9125:
9103:
9067:
9053:
9035:
9012:
8985:{\displaystyle {\mathfrak {a}}_{i}}
8971:
8954:{\displaystyle {\mathfrak {a}}_{i}}
8940:
8923:{\displaystyle {\mathfrak {a}}_{i}}
8909:
8892:{\textstyle R_{i}\to R=\prod R_{i}}
8843:{\displaystyle R_{i},1\leq i\leq n}
8775:
8744:
8714:
8665:
8615:
8526:
8509:
8478:
8455:
7660:is an abelian group under addition.
6839:is the set of all automorphisms of
6718:("quotient ring" is defined below).
6171:Like a group, a ring is said to be
5703:, and those rings are said to have
5325:the addition and multiplication of
4287:It is a subring of the ring of all
4036:Associative algebra Β§ Examples
3543:, the square matrices of dimension
2762:may be considered as an element of
1858:, along with the two operations of
1112:Β§ Variations on the definition
24:
21542:, 2nd ed., Vol. 2. Cambridge, MA:
21384:
20766:Exercises in classical ring theory
18211:Every ring can be thought of as a
17911:
17771:
16297:-algebra. The matrix ring of size
16114:
15737:is algebraically closed, then all
15452:is a unique factorization domain,
15165:(It is also the same thing as the
15161:in variables that are elements of
13235:-module, then the localization of
11437:
11142:
10963:
10761:
10614:is a product of linear factors in
10455:be the ring of all functions from
9920:. Because of this, the polynomial
9388:
6783:is a ring homomorphism, called an
6653:) is a homomorphism from the ring
5904:, that is, the set of finite sums
5245:. The set of units of a ring is a
4802:
4797:
3190:
3055:
2562:is the remainder when the integer
2323:is the remainder when the integer
2018:
527:
25:
21655:
20879:"A primer of commutative algebra"
19355:, which for any commutative ring
18854:{\displaystyle \{0,1,2,\ldots \}}
18725:
18694:-group, it is meant a group with
18261:on an abelian group is simply an
18177:Face ring of a simplicial complex
17322:It also comes with the valuation
17213:be the set of all functions from
16933:
16545:form an abelian group called the
15998:is semisimple if and only if the
15835:principal ideal domains
15780:. A regular local ring is a UFD.
14920:{\displaystyle \mathbb {Z} _{p}.}
14831:{\displaystyle \mathbb {Q} _{p}.}
14336:{\displaystyle \mathbb {Z} _{p}.}
13340:a multiplicatively closed subset
13182:{\displaystyle R_{\mathfrak {p}}}
13153:{\displaystyle R/{\mathfrak {p}}}
13128:, then the field of fractions of
13039:{\displaystyle R_{\mathfrak {p}}}
12967:{\displaystyle R_{\mathfrak {p}}}
12752:consists of elements of the form
12315:{\displaystyle \varprojlim R_{i}}
11595:be a sequence of rings such that
10838:Matrix ring and endomorphism ring
8431:{\textstyle \prod _{i\in I}R_{i}}
8005:
7903:The axioms of modules imply that
7839:are defined similarly by writing
7348:is a ring homomorphism such that
5776:. More generally, given a subset
4873:
4402:of sets and multiplication to be
4029:
3486:
3422:
3355:
3294:
3233:
594:Noncommutative algebraic geometry
20905:van der Waerden, Bartel Leendert
20414:, New York: Dover Publications,
20079:"Associative rings and algebras"
19847:"The development of Ring Theory"
19392:
19275:
19272:
19269:
19266:
19243:
19240:
19237:
19234:
19231:
19196:
19193:
19190:
19187:
17505:{\displaystyle +:R\times R\to R}
16659:{\displaystyle \mathbb {Q} _{p}}
16582:quasi-algebraically closed field
16289:-algebra is a field, any simple
16170:is a positive integer, and each
16055:, the following are equivalent:
15859:
15853:algebraically closed fields
14864:{\displaystyle \mathbb {Q} _{p}}
14612:into prime numbers (we also put
14230:{\displaystyle R\to {\hat {R}}.}
13861:{\displaystyle {\mathfrak {m}},}
12888:{\displaystyle R\left=R/(tf-1).}
11893:
11860:
11662:{\displaystyle \varinjlim R_{i}}
10741:consists of formal power series
9647:of all polynomials in variables
8000:
7044:is analogous to the notion of a
7029:
6182:of left ideals is called a left
5439:of integers is a subring of the
4682:{\displaystyle (\mathbb {N} ,+)}
4363:{\displaystyle \mathbb {Z} _{p}}
2834:{\displaystyle {\overline {x}},}
2754:is an integer, the remainder of
2027:The additive identity is unique.
1735:
1704:is often omitted, in which case
491:{\displaystyle \mathbb {Q} _{p}}
454:{\displaystyle \mathbb {Z} _{p}}
21445:"Idealtheorie in Ringbereichen"
21268:"Why all rings should have a 1"
21172:The Art of Computer Programming
20647:(1943). "The Theory of Rings".
20371:
20071:
19687:
19678:
19665:
19486:Algebra over a commutative ring
19412:together with a multiplication
19210:through the category of rings:
18307:is the set of all morphisms of
18257:). The monoid action of a ring
17852:one can associate its integral
16285:. Since the center of a simple
15718:then such a cyclic module (for
15401:.) The fundamental question in
14072:be a commutative ring, and let
13841:is exact for any maximal ideal
13117:{\displaystyle {\mathfrak {p}}}
12431:
12424:
11832:{\displaystyle R=\varinjlim R.}
11166:
10789:
10570:{\displaystyle {\overline {f}}}
10512:
10463:be the identity function. Each
9850:
9580:as a subring. It is called the
9203:
9099:
9049:
8899:the inclusions with the images
8680:
7609:to every pair of an element of
7528:that gives an isomorphism from
6937:An algebra homomorphism from a
6886:and a unique ring homomorphism
6439:the following identities hold:
6374:
5963:
5734:commutes with every element in
5722:denote the set of all elements
5184:is necessarily a zero divisor.
4736:The natural numbers (including
4063:is itself a ring as well as an
4058:algebra over a commutative ring
3896:
2865:{\displaystyle {\overline {x}}}
2240:with the following operations:
1845:
1196:rings of differential operators
20954:Wilder, Raymond Louis (1965).
20631:(1964). "Structure of rings".
20609:Hungerford, Thomas W. (1997).
19761:
19729:Mac Lane & Birkhoff (1967)
19646:
19629:
19434:
19294:
19251:
19227:
19183:
19167:
19155:
19099:
19059:
19018:
18978:
18931:
18478:
18475:
18469:
18463:
18454:
18448:
18445:
18433:
18423:
18417:
18408:
18402:
18393:
18387:
18384:
18372:
18201:Category-theoretic description
17997:
17983:
17940:
17926:
17889:
17875:
17758:
17752:
17687:
17681:
17662:
17656:
17625:
17613:
17496:
17474:which makes the addition map (
17306:
17294:
17288:
17282:
17257:
17251:
17248:
17236:
17200:
17196:
17190:
17186:
17168:More generally, given a field
17150:
17146:
17140:
17136:
17083:
17079:
17073:
17069:
16891:
16877:
16818:{\displaystyle A\otimes _{k}F}
16761:{\displaystyle -\otimes _{k}F}
16689:
16683:
16609:
16601:
16503:
16497:
16494:
16488:
16227:{\displaystyle A\otimes _{k}F}
16146:
16133:
15683:
15677:
15609:
15603:
15568:{\displaystyle t\cdot v=f(v),}
15559:
15553:
15407:ring of (generalized) integers
15405:is on the extent to which the
15317:
15305:
15302:
15290:
15272:{\displaystyle A\otimes _{R}B}
15190:
14728:
14719:
14703:
14694:
14680:
14663:
14629:
14620:
14591:
14585:
14553:
14547:
14516:
14507:
14433:
14425:
14421:
14218:
14209:
14107:
13906:
13895:
13889:
13878:
13825:
13807:
13792:
13774:
13741:
13730:
13724:
13713:
13646:
13614:
13587:
13555:
13357:
13209:
13199:
12879:
12864:
12856:
12850:
12505:
12482:
12466:
12129:of the structure sheaf at the
12102:
12096:
12046:
12042:
12036:
12032:
12006:
12002:
11996:
11992:
11957:
11953:
11947:
11943:
11823:
11778:
11756:
11724:
11488:
11485:
11472:
11456:
11409:
11403:
11189:
11173:
11170:
11160:
11154:
11138:
11135:
11114:
11028:
10981:
10975:
10956:
10943:
10728:
10724:
10718:
10714:
10516:
10503:
10500:
10494:
10394:
10388:
10355:
10352:
10346:
10299:
10098:
10088:
10082:
10076:
10073:
10061:
9959:
9953:
9947:
9866:
9860:
9854:
9841:
9838:
9832:
9426:
9420:
8867:
8787:
8731:
8728:
8546:
8540:
8296:
8244:
8238:
8212:
8206:
8180:
8170:
8118:
8112:
8086:
8080:
8054:
7807:
7798:
7786:
7777:
7729:
7717:
7689:
7677:
7512:
7390:
7238:
7229:
7223:
7211:
7208:
7196:
7177:
7165:
7159:
7147:
7141:
7129:
6914:
6868:
6751:
6739:
6576:
6563:
6552:
6546:
6537:
6531:
6522:
6510:
6499:
6493:
6484:
6478:
6469:
6457:
5788:be the set of all elements in
5631:the subring generated by
5466:
5460:
4926:
4894:
4861:is a rng, but not a ring: the
4834:
4822:
4816:
4810:
4783:
4777:
4774:
4762:
4676:
4662:
4608:ring of differential operators
4138:
4135:
4103:
4100:
3208:
3202:
3073:
3067:
2845:. The additive inverse of any
1287:under addition, meaning that:
685:need not exist. Informally, a
532:
519:
13:
1:
21556:American Mathematical Monthly
21526:American Mathematical Society
21064:Gilmer, R.; Mott, J. (1973).
21034:. Cambridge University Press.
20792:Graduate Texts in Mathematics
20722:Lectures on modules and rings
20383:
20277:Atiyah & Macdonald (1969)
20096:Gardner & Wiegandt (2003)
18032:universal coefficient theorem
17512:) and the multiplication map
17423:matrices over the real field
16849:is canonically isomorphic to
16043:
15242:. Then the tensor product of
15079:{\displaystyle \mathbb {Z} ,}
14372:{\displaystyle \mathbb {Q} .}
14063:
11515:(cf. below) is of this form.
9965:{\displaystyle f\mapsto f(x)}
9782:is a principal ideal domain.
7936:is a ring homomorphism, then
7560:generalizes the concept of a
7112:together with the operations
6951:representation of the algebra
5589:{\displaystyle 2\mathbb {Z} }
5560:{\displaystyle \mathbb {Z} ;}
5527:{\displaystyle 2\mathbb {Z} }
4947:, one can define the product
4883:For each nonnegative integer
4727:{\displaystyle \mathbb {Z} .}
4593:commute with the elements of
4278:{\displaystyle \mathbb {Q} .}
4196:-algebra. The operations are
3965:Idealtheorie in Ringbereichen
3535:More generally, for any ring
2741:{\displaystyle \mathbb {Z} .}
1890:{\displaystyle \mathbb {Z} ,}
1256:
677:: multiplication need not be
21066:"Associative Rings of Order"
20967:. Vol. 1. Van Nostrand.
20854:Matsumura, Hideyuki (1989).
20845:; Birkhoff, Garrett (1967).
19704:
18652:gives rise to a morphism of
18609:be the set of all morphisms
18564:gives rise to a morphism of
18287:be an abelian group and let
18246:{\displaystyle \mathbb {Z} }
18133:, which is more or less the
18050:-multilinear form to get a (
17726:{\displaystyle \mathbb {Z} }
17440:{\displaystyle \mathbb {R} }
16792:. It consists of such that
16326:Two central simple algebras
15783:The following is a chain of
15039:, then the quotient ring of
14791:{\displaystyle \mathbb {Q} }
14765:{\displaystyle \mathbb {Q} }
14490:{\displaystyle \mathbb {R} }
14464:{\displaystyle \mathbb {Q} }
14440:{\displaystyle x\mapsto |x|}
14402:{\displaystyle \mathbb {Q} }
14298:generated by a prime number
14281:{\displaystyle \mathbb {Z} }
14255:Krull's intersection theorem
13100:at the prime ideal zero. If
12108:{\displaystyle \varinjlim k}
11864:
11584:Limits and colimits of rings
10562:
10524:
10425:
10383:
10335:
10308:{\displaystyle \phi :R\to S}
9804:{\displaystyle R\subseteq S}
9285:are central idempotents and
7487:
7424:
7371:
6725:is a unit element in a ring
6670:{\displaystyle \mathbb {Z} }
6630:), or equivalently if it is
5498:{\displaystyle \mathbb {Z} }
5472:{\displaystyle \mathbb {Z} }
5430:{\displaystyle \mathbb {Z} }
4644:{\displaystyle \mathbb {N} }
4618:
4258:in a quadratic extension of
4249:{\displaystyle \mathbb {Z} }
4187:{\displaystyle \mathbb {R} }
3008:
2995:
2977:
2944:
2926:
2857:
2823:
2661:
2648:
2635:
2608:
2595:
2582:
2508:
2495:
2463:
2450:
2437:
2410:
2397:
2384:
2269:
2256:
2220:
2207:
2194:
2181:
1759:
1724:Variations on the definition
200:{\displaystyle \mathbb {Z} }
7:
21332:Weibel, Charles A. (2013),
21312:Springer, Tonny A. (1977),
21242:Pierce, Richard S. (1982).
20693:University of Chicago Press
20549:Gallian, Joseph A. (2006).
20440:Lie algebras and Lie groups
20219:, Ch. XVII. Proposition 1.1
20180:, p. 122, Theorem 2.10
20083:Encyclopedia of Mathematics
19984:Zariski & Samuel (1958)
19773:Encyclopedia of Mathematics
19526:Simplicial commutative ring
19478:
18761:
18617:, having the property that
18018:, for which the methods of
17395:A ring may be viewed as an
17349:(which makes sense even if
16780:. Its kernel is denoted by
15938:
15905:, is a generalization of a
15882:Wedderburn's little theorem
15391:unique factorization domain
14302:; it is called the ring of
13545:The localization is exact:
12535:{\displaystyle R\to R\left}
12219:are all the identities and
12139:finitely generated subrings
12125:(more succinctly it is the
11849:of the same characteristic
9756:is a Noetherian ring, then
8391:are rings indexed by a set
7894:dimension of a vector space
7605:(associating an element of
6329:is prime if for any ideals
5180:. A nilpotent element in a
3993:Encyclopedia of Mathematics
3585:
2351:, this remainder is either
1794:, to simplify terminology.
1774:need not necessarily equal
352:Unique factorization domain
10:
21660:
21514:A Survey of Modern Algebra
21105:Algebra: A Graduate Course
20890:(2nd ed.), Springer,
20860:Cambridge University Press
20828:(3rd ed.), Springer,
20391:Garling, D. J. H. (2022).
20357:, p. 162, Theorem 3.2
18886:be a category with finite
18817:the non-negative integers
18709:Any ring can be seen as a
18656:: right multiplication by
18221:category of abelian groups
18204:
17391:Rings with extra structure
17379:
17206:{\displaystyle k(\!(G)\!)}
17089:{\displaystyle k(\!(t)\!)}
16937:
16735:, then the base extension
16262:
15916:
15368:
15355:Tensor product of algebras
15352:
15059:as a base ring instead of
15055:. If we used a ring, say,
15043:by the ideal generated by
13936:localization of a category
13336:is a commutative ring and
12703:is a commutative ring and
10841:
9776:is a field if and only if
9588:. More generally, the set
9392:
9381:of a family of rings (cf.
8445:be a commutative ring and
8009:
7549:
7451:For any ring homomorphism
7033:
6957:Given a ring homomorphism
6814:is a ring endomorphism of
6378:
5831:
5643:, the smallest subring of
5613:, the smallest subring of
5302:
5298:
4695:(not all the elements are
4406:. This is an example of a
4232:, the integral closure of
4033:
3609:
3567:Ring theory Β§ History
3564:
3560:
2136:
2133:Example: Integers modulo 4
1766:Although ring addition is
112:Tensor product of algebras
29:
21409:10.1515/crll.1915.145.139
19299:In algebraic geometry, a
18878:Ring object in a category
18644:. It was seen that every
17572:together with operations
17407:is a ring that is also a
17100:comes with the valuation
16281:and is simple if it is a
15892:CartanβBrauerβHua theorem
14648:{\displaystyle |0|_{p}=0}
14245:is a proper ideal, or if
13950:with respect to a subset
13538:running over elements in
12602:uniquely factors through
12121:of nonempty open subsets
10629:Hilbert's Nullstellensatz
9924:is often also denoted by
9762:is a Noetherian ring. If
9407:, the set of polynomials
8571:Chinese remainder theorem
7617:) that satisfies certain
7545:
3578:, one of the founders of
2084:is a unit element), then
1089:such that the ring is an
21019:Ann. Soc. Sci. Bruxelles
20463:(2nd ed.). Pearson.
20442:(2nd ed.), Springer
19912:, p. 86, footnote 1
19719:, p. 96, Ch 1, Β§8.1
19622:
19532:Special types of rings:
19085:(additive inverse), and
18910:equipped with morphisms
18137:given a ring structure.
18083:Burnside ring of a group
17829:Many different kinds of
17353:is not commutative). If
17125:formal power series ring
17119:; the valuation ring of
16731:is a field extension of
16563:ArtinβWedderburn theorem
16234:is semisimple for every
16183:ArtinβWedderburn theorem
14604:denotes the exponent of
14597:{\displaystyle v_{p}(n)}
14385:-adic absolute value on
14094:is the projective limit
12711:, then the localization
12021:formal power series ring
11509:ArtinβWedderburn theorem
10900:composition of functions
10703:formal power series ring
10451:To give an example, let
7475:produces a homomorphism
6645:to its remainder modulo
6234:{\displaystyle x,y\in R}
6192:HopkinsβLevitzki theorem
5827:
5210:is an element such that
4428:, the set of all square
3029:Example: 2-by-2 matrices
2103:itself is the zero ring.
1151:affine algebraic variety
390:Formal power series ring
340:Integrally closed domain
21594:Elemente der Mathematik
21486:Handbook of Mathematics
20910:Moderne Algebra. Teil I
20886:Rotman, Joseph (1998),
20856:Commutative Ring Theory
20570:Radical Theory of Rings
20501:Algebra I, Chapters 1β3
19616:discrete valuation ring
19506:Non-associative algebra
19501:Glossary of ring theory
18873:Other ring-like objects
18191:algebraic combinatorics
17466:if its set of elements
17455:as a real vector space.
17365:th coefficient is
15403:algebraic number theory
14963:Cohen structure theorem
14873:consisting of elements
14290:at the principal ideal
14192:{\displaystyle R/I^{n}}
13443:{\displaystyle R\left.}
13096:is the localization of
13011:{\displaystyle R\left.}
12639:{\displaystyle R\left.}
12587:{\displaystyle R\left,}
11703:for sufficiently large
11270:is a division ring. If
9935:. The image of the map
8012:Direct product of rings
7944:by the multiplication:
6945:of a vector space over
6318:{\displaystyle y\in P.}
6263:{\displaystyle xy\in P}
5808:(or commutant) of
5674:times) can be zero. If
5409:is a ring homomorphism.
4424:and any natural number
4372:over all prime numbers
3917:and so on; in general,
2347:is always smaller than
1694:(right distributivity).
1580:multiplicative identity
1132:algebraic number theory
1126:, is a major branch of
683:multiplicative inverses
399:Algebraic number theory
92:Total ring of fractions
21617:van der Waerden, B. L.
21083:10.3792/pja/1195519146
20412:Advanced Number Theory
19973:van der Waerden (1930)
19444:
19347:
19286:
19204:
19119:
19079:
19038:
18998:
18951:
18898:(an empty product). A
18855:
18733:
18595:. In other words, let
18492:
18247:
18065:characteristic classes
18004:
17950:
17915:
17793:
17727:
17697:
17645:
17568:is a commutative ring
17506:
17441:
17361:with the series whose
17316:
17207:
17160:
17090:
16919:
16825:is a matrix ring over
16819:
16762:
16714:
16660:
16616:
16539:
16538:{\displaystyle =\left}
16475:
16409:
16338:if there are integers
16321:SkolemβNoether theorem
16265:Central simple algebra
16228:
16201:if the base extension
16153:
16112:
15955:For any division ring
15841:Euclidean domains
15805:commutative rings
15773:
15712:
15654:
15569:
15523:
15364:Special kinds of rings
15345:
15273:
15205:
15151:
15080:
14921:
14893:is isomorphic to
14865:
14832:
14792:
14774:and the completion of
14766:
14742:
14649:
14598:
14562:
14491:
14465:
14441:
14403:
14373:
14337:
14282:
14231:
14199:induce a homomorphism
14193:
14154:
14054:
14053:{\displaystyle R\left}
14009:
14008:{\displaystyle R\left}
13916:
13862:
13835:
13751:
13697:
13696:{\displaystyle R\left}
13656:
13529:
13444:
13392:
13324:
13219:
13183:
13154:
13118:
13086:
13040:
13012:
12968:
12939:
12889:
12807:
12746:
12745:{\displaystyle R\left}
12681:
12680:{\displaystyle R\left}
12640:
12588:
12536:
12489:
12456:, there exists a ring
12348:
12316:
12109:
12056:
12013:
11917:
11833:
11710:Examples of colimits:
11663:
11498:
11435:
11374:
11373:{\displaystyle U_{i},}
11333:
11303:
11208:Any ring homomorphism
11199:
11053:
10991:
10810:
10765:
10735:
10684:
10683:{\displaystyle k\left}
10602:, there exists a ring
10571:
10533:
10434:
10407:
10365:
10309:
10144:
10042:
10041:{\displaystyle k\left}
9966:
9873:
9805:
9725:
9724:{\displaystyle R\left}
9687:
9641:
9640:{\displaystyle R\left}
9567:
9353:
9272:
9238:
9140:
8986:
8955:
8924:
8893:
8844:
8797:
8710:
8652:
8610:
8553:
8493:
8432:
8307:
7940:is a left module over
7818:
7522:
7442:
7400:
7258:
6926:
6878:
6843:whose restrictions to
6820:Frobenius homomorphism
6777:
6710:
6671:
6600:
6319:
6290:
6289:{\displaystyle x\in P}
6264:
6235:
6142:. The principal ideal
6028:
5663:. It is possible that
5649:characteristic subring
5590:
5561:
5528:
5499:
5473:
5431:
5413:For example, the ring
5237:multiplicative inverse
5204:
5002:
4987:
4933:
4851:
4728:
4683:
4645:
4517:, then the set of all
4364:
4327:
4279:
4250:
4188:
4145:
3957:multiplicative inverse
3908:
3582:
3526:
3459:
3395:
3331:
3270:
3215:
3169:
3020:
2956:
2903:
2866:
2835:
2796:
2742:
2713:
2673:
2617:
2554:
2517:
2475:
2419:
2315:
2278:
2234:
2006:
1891:
1867:
1803:multiplicative inverse
1642:(left distributivity).
556:Noncommutative algebra
539:
492:
455:
407:Algebraic number field
358:Principal ideal domain
247:
201:
139:Frobenius endomorphism
21608:10.1007/s000170050029
21478:Historical references
21132:Annals of Mathematics
20935:Warner, Seth (1965).
20826:Undergraduate algebra
20410:Cohn, Harvey (1980),
19957:, p. 144, axiom
19924:, p. 144, axiom
19445:
19348:
19287:
19205:
19120:
19080:
19044:(additive identity),
19039:
18999:
18952:
18856:
18668:, as a function from
18493:
18248:
18022:are not well-suited.
18005:
17951:
17895:
17801:binomial coefficients
17794:
17728:
17698:
17631:
17507:
17442:
17317:
17208:
17161:
17159:{\displaystyle k\!].}
17091:
17058:formal Laurent series
16920:
16820:
16763:
16715:
16661:
16617:
16540:
16476:
16410:
16229:
16154:
16092:
15959:and positive integer
15811:integral domains
15771:
15759:Jordan canonical form
15713:
15655:
15570:
15524:
15377:ring with no nonzero
15346:
15274:
15206:
15152:
15081:
14994:of symbols, that is,
14938:is the completion of
14922:
14866:
14833:
14793:
14767:
14743:
14650:
14599:
14563:
14492:
14466:
14442:
14404:
14374:
14338:
14283:
14232:
14194:
14155:
14055:
14010:
13962:from the category of
13917:
13863:
13836:
13752:
13698:
13657:
13530:
13445:
13393:
13325:
13220:
13184:
13155:
13119:
13087:
13041:
13013:
12969:
12940:
12890:
12808:
12747:
12682:
12641:
12589:
12537:
12490:
12349:
12317:
12110:
12057:
12055:{\displaystyle k\!].}
12014:
11927:formal Laurent series
11918:
11834:
11664:
11618:. Then the union (or
11499:
11415:
11375:
11334:
11283:
11200:
11054:
10992:
10811:
10751:
10736:
10685:
10572:
10534:
10435:
10408:
10366:
10310:
10145:
10043:
9967:
9874:
9806:
9726:
9688:
9642:
9568:
9354:
9273:
9244:By the conditions on
9239:
9141:
8987:
8956:
8925:
8894:
8845:
8798:
8690:
8632:
8590:
8554:
8494:
8433:
8308:
7966:is commutative or if
7819:
7570:scalar multiplication
7523:
7443:
7401:
7259:
6927:
6879:
6829:of a field extension
6778:
6711:
6679:to the quotient ring
6672:
6601:
6320:
6291:
6265:
6236:
6029:
5850:is a nonempty subset
5800:is a subring of
5764:is a subring of
5668:Β· 1 = 1 + 1 + ... + 1
5591:
5562:
5529:
5500:
5474:
5432:
5205:
5003:
4967:
4934:
4852:
4729:
4684:
4646:
4365:
4328:
4280:
4251:
4189:
4155:with coefficients in
4146:
4083:with coefficients in
4056:A unital associative
3909:
3574:
3527:
3460:
3396:
3332:
3271:
3216:
3180:matrix multiplication
3170:
3021:
2957:
2904:
2867:
2836:
2797:
2743:
2714:
2674:
2618:
2555:
2518:
2476:
2420:
2316:
2279:
2235:
2007:
1892:
1853:
1184:representation theory
540:
493:
456:
248:
202:
21639:Algebraic structures
21621:A History of Algebra
21397:J. Reine Angew. Math
21272:Mathematics Magazine
21245:Associative algebras
21206:"Class field theory"
20824:Lang, Serge (2005),
20691:(Revised ed.),
20592:Noncommutative rings
20553:. Houghton Mifflin.
20428:Serre, J-P. (1950),
19416:
19329:
19214:
19133:
19089:
19048:
19008:
18961:
18914:
18821:
18711:preadditive category
18362:
18235:
18195:simplicial polytopes
18187:StanleyβReisner ring
18060:)-multilinear form.
17970:
17862:
17831:mathematical objects
17814:totally ordered ring
17805:RiemannβRoch theorem
17739:
17715:
17600:
17478:
17429:
17233:
17180:
17130:
17063:
16853:
16796:
16739:
16674:
16641:
16630:is a nonarchimedean
16624:theorem of Frobenius
16592:
16485:
16419:
16350:
16205:
16181:is a division ring (
16089:
16006:does not divide the
15729:is represented by a
15664:
15597:
15535:
15460:
15287:
15250:
15184:
15096:
15065:
14899:
14846:
14810:
14780:
14754:
14659:
14616:
14572:
14503:
14479:
14453:
14415:
14391:
14358:
14350:-adic absolute value
14315:
14270:
14261:is a maximal ideal.
14203:
14168:
14098:
14076:be an ideal of
14023:
13978:
13872:
13845:
13768:
13707:
13666:
13549:
13455:
13410:
13347:
13250:
13193:
13164:
13132:
13104:
13057:
13021:
12978:
12949:
12910:
12817:
12756:
12715:
12650:
12606:
12554:
12550:to unit elements in
12499:
12460:
12326:
12286:
12077:
12026:
11937:
11853:
11718:
11633:
11544:of right modules of
11522:and the matrix ring
11387:
11354:
11274:
11098:
11001:
10927:
10748:
10734:{\displaystyle R\!]}
10708:
10635:
10554:
10488:
10417:
10375:
10327:
10287:
10272:, the derivative of
10055:
9999:
9941:
9826:
9789:
9697:
9651:
9592:
9414:
9313:
9248:
9162:
9000:
8965:
8934:
8903:
8854:
8809:
8577:
8503:
8499:be ideals such that
8449:
8399:
8044:
7977:is contained in the
7831:these axioms define
7666:
7552:Module (mathematics)
7479:
7410:
7363:
7119:
6987:. The image of
6943:endomorphism algebra
6906:
6860:
6733:
6685:
6659:
6612:A ring homomorphism
6446:
6300:
6274:
6245:
6213:
6209:if for any elements
5911:
5575:
5546:
5513:
5487:
5452:
5419:
5293:general linear group
5291:, and is called the
5194:
4951:
4891:
4863:Dirac delta function
4759:
4713:
4659:
4633:
4415:Noncommutative rings
4400:symmetric difference
4345:
4303:
4264:
4238:
4176:
4094:
3943:Fraenkel and Noether
3641:
3469:
3405:
3341:
3280:
3225:
3186:
3051:
2966:
2915:
2878:
2849:
2815:
2768:
2727:
2688:
2627:
2574:
2529:
2487:
2429:
2376:
2290:
2248:
2147:
2080:(or more generally,
1909:
1876:
1537:There is an element
1388:There is an element
962:Group with operators
905:Complemented lattice
740:Algebraic structures
671:algebraic structures
562:Noncommutative rings
511:
473:
436:
280:Non-associative ring
216:
189:
146:Algebraic structures
21520:Faith, Carl (1999)
21366:Commutative algebra
20965:Commutative Algebra
20639:(Revised ed.).
20489:Algèbre commutative
20322:, Ch 1, Theorem 3.8
20231:, Proposition 1.3.1
20192:, Ch 5. Β§1, Lemma 2
19695:centrally primitive
18752:nonassociative ring
18746:Nonassociative ring
18664:, to a morphism of
18171:Birational geometry
18167:commutative algebra
18165:makes heavy use of
18145:To any irreducible
18126:representation ring
18106:representation ring
18073:algebraic varieties
17405:associative algebra
16965:such that, for any
16305:will be denoted by
16253:separable extension
15977:is semisimple (and
15948:is semisimple (and
15744:'s are of the form
15642:
15589:is a direct sum of
15515:
15490:
13824:
13791:
13757:is exact over
11339:is a direct sum of
11328:
11131:
11045:
11024:
10579:polynomial function
9152:central idempotents
9118:
8024:be rings. Then the
6041:is a left ideal if
5834:Ideal (ring theory)
5705:characteristic zero
5629:, and it is called
5332:to give operations
5281:over a field, then
5224:in linear algebra.
4887:, given a sequence
4879:Products and powers
4806:
4614:are noncommutative.
4390:is a set, then the
4153:formal power series
1192:functional analysis
1120:Commutative algebra
1016:Composition algebra
776:Quasigroup and loop
321:Commutative algebra
160:Associative algebra
42:Algebraic structure
21510:Mac Lane, Saunders
21461:10.1007/bf01464225
21301:Serre, Jean-Pierre
20974:Special references
20877:Milne, J. (2012).
20843:Mac Lane, Saunders
20449:General references
20267:, end of Chapter 7
19900:, pp. 143β145
19521:Spectrum of a ring
19440:
19399:algebraic topology
19343:
19282:
19200:
19115:
19075:
19034:
19004:(multiplication),
18994:
18947:
18890:. Let pt denote a
18851:
18488:
18486:
18268:. Essentially, an
18243:
18229:tensor product of
18183:simplicial complex
18163:algebraic geometry
18149:is associated its
18135:Grothendieck group
18124:is associated its
18101:free abelian group
18091:is associated its
18020:point-set topology
18000:
17946:
17789:
17723:
17693:
17555:Euclidean topology
17502:
17437:
17312:
17278:
17203:
17156:
17086:
17030:is the subring of
16915:
16815:
16758:
16710:
16656:
16612:
16553:and is denoted by
16535:
16471:
16405:
16224:
16149:
16032:over a field is a
15963:, the matrix ring
15907:quaternion algebra
15778:regular local ring
15774:
15708:
15650:
15621:
15565:
15519:
15494:
15469:
15341:
15269:
15201:
15147:
15092:. For example, if
15076:
14917:
14861:
14828:
14788:
14762:
14738:
14645:
14594:
14558:
14487:
14461:
14437:
14399:
14369:
14333:
14278:
14227:
14189:
14150:
14124:
14050:
14005:
13912:
13858:
13831:
13810:
13777:
13747:
13693:
13652:
13525:
13493:
13440:
13388:
13320:
13215:
13189:and is denoted by
13179:
13150:
13114:
13082:
13036:
13008:
12964:
12935:
12885:
12803:
12742:
12699:. For example, if
12677:
12636:
12584:
12532:
12485:
12442:field of fractions
12344:
12343:
12322:is the subring of
12312:
12297:
12105:
12088:
12052:
12009:
11971:
11913:
11886:
11829:
11770:
11659:
11644:
11494:
11370:
11346:-copies of simple
11329:
11304:
11252:is a simple right
11195:
11117:
11063:-linear map, then
11049:
11031:
11010:
10987:
10906:and is denoted by
10894:-linear maps from
10868:and is denoted by
10806:
10731:
10680:
10567:
10529:
10430:
10403:
10361:
10305:
10140:
10038:
9962:
9869:
9801:
9750:rational functions
9721:
9683:
9637:
9563:
9379:restricted product
9349:
9268:
9234:
9136:
9100:
8982:
8951:
8920:
8889:
8840:
8793:
8725:
8627:
8549:
8489:
8428:
8417:
8303:
8301:
7814:
7812:
7613:and an element of
7558:module over a ring
7518:
7438:
7396:
7254:
7252:
6934:is an epimorphism.
6922:
6874:
6785:inner automorphism
6773:
6706:
6667:
6596:
6594:
6315:
6286:
6260:
6231:
6024:
5858:such that for any
5586:
5557:
5524:
5495:
5469:
5427:
5273:. For example, if
5200:
5120:Elements in a ring
4998:
4929:
4847:
4789:
4724:
4679:
4641:
4439:with entries from
4360:
4323:
4296:profinite integers
4289:algebraic integers
4275:
4246:
4230:quadratic integers
4207:be a set, and let
4184:
4144:{\displaystyle R]}
4141:
3904:
3902:
3596:algebraic integers
3594:and the theory of
3583:
3522:
3513:
3512:
3455:
3449:
3448:
3391:
3382:
3381:
3327:
3321:
3320:
3266:
3260:
3259:
3211:
3165:
3117:
3037:with entries in a
3033:The set of 2-by-2
3016:
2952:
2899:
2862:
2831:
2792:
2738:
2709:
2669:
2613:
2550:
2513:
2471:
2415:
2311:
2274:
2230:
2139:Modular arithmetic
2002:
1899:consisting of the
1887:
1868:
1587:Multiplication is
1239:algebraic geometry
1136:algebraic geometry
1107:ring with identity
693:equipped with two
575:Semiprimitive ring
535:
488:
451:
259:Related structures
243:
197:
133:Inner automorphism
119:Ring homomorphisms
21623:, Springer-Verlag
21587:(February 1998).
21506:Birkhoff, Garrett
21227:978-0-88275-228-0
21214:Nagata, Masayoshi
21118:978-0-8218-4799-2
21040:Cohn, Paul Moritz
21010:978-0-13-155623-2
20992:978-0-13-155615-7
20920:978-3-540-56799-8
20869:978-0-521-36764-6
20801:978-0-387-95385-4
20688:Commutative rings
20683:Kaplansky, Irving
20674:978-0-486-47189-1
20522:978-1-85233-587-8
20509:Cohn, Paul Moritz
20479:. AddisonβWesley.
20473:Macdonald, Ian G.
20421:978-0-486-64023-5
20402:978-1-108-83892-4
20377:Serre, p. 44
20255:, Proposition 6.4
20043:Hungerford (1997)
19548:Differential ring
19496:Category of rings
19473:symmetric spectra
19359:returns the ring
19262:
19259:
19108:
19068:
19027:
18987:
18940:
18866:tropical semiring
18716:Additive functors
18583:. Consider those
18297:endomorphism ring
18225:monoidal category
18223:(thought of as a
18207:Category of rings
18147:algebraic variety
18077:Schubert calculus
18024:Cohomology groups
17963:. There are also
17847:topological space
17816:is a ring with a
17778:
17735:is a Ξ»-ring with
17263:
16022:Clifford algebras
16016:Maschke's theorem
15992:, the group ring
15988:and finite group
15926:semisimple module
15919:Semisimple module
15733:. Thus, if, say,
15217:category of rings
15213:forgetful functor
15196:
15167:symmetric algebra
14221:
14164:to the quotients
14117:
14110:
13486:
12945:one often writes
12488:{\displaystyle R}
12425:Β§ Completion
12290:
12081:
11964:
11879:
11867:
11843:algebraic closure
11763:
11637:
11538:Morita equivalent
10890:, the set of all
10848:Endomorphism ring
10565:
10527:
10446:symmetric algebra
10428:
10386:
10338:
10197:is an element in
8402:
7827:When the ring is
7588:equipped with an
7556:The concept of a
7490:
7427:
7374:
7291:is also called a
6847:are the identity.
6616:is said to be an
6381:Ring homomorphism
6001:
5976:
5968:
5692:. In some rings,
5203:{\displaystyle e}
5156:nilpotent element
5008:recursively: let
4490:endomorphism ring
4488:form a ring, the
4317:
4070:. Some examples:
4041:Commutative rings
3891:
3551:form a ring; see
3129:
3011:
2998:
2980:
2947:
2929:
2860:
2826:
2664:
2651:
2638:
2611:
2598:
2585:
2511:
2498:
2466:
2453:
2440:
2413:
2400:
2387:
2272:
2259:
2223:
2210:
2197:
2184:
2056:absorbing element
1783:commutative rings
1420:additive identity
1188:operator algebras
1124:commutative rings
1075:
1074:
695:binary operations
659:
658:
616:Geometric algebra
327:Commutative rings
178:Category of rings
16:(Redirected from
21651:
21624:
21612:
21610:
21580:
21517:
21472:
21436:
21420:
21403:(145): 139β176.
21379:
21348:
21328:
21315:Invariant theory
21308:
21296:
21287:
21259:
21238:
21209:
21200:
21175:
21163:
21127:Jacobson, Nathan
21122:
21096:
21087:
21085:
21070:Proc. Japan Acad
21060:
21049:
21035:
21026:
21013:
20995:
20968:
20959:
20950:
20931:
20900:
20882:
20873:
20850:
20838:
20820:
20779:
20757:
20735:
20713:
20678:
20661:Jacobson, Nathan
20656:
20645:Jacobson, Nathan
20640:
20629:Jacobson, Nathan
20624:
20605:
20583:
20564:
20545:
20525:
20504:
20492:
20480:
20464:
20443:
20434:
20424:
20406:
20378:
20375:
20369:
20364:
20358:
20352:
20346:
20341:
20335:
20329:
20323:
20313:
20307:
20301:
20295:
20286:
20280:
20274:
20268:
20262:
20256:
20250:
20244:
20238:
20232:
20226:
20220:
20214:
20208:
20202:
20193:
20187:
20181:
20175:
20169:
20163:
20157:
20151:
20145:
20139:
20133:
20127:
20121:
20115:
20109:
20104:
20098:
20093:
20087:
20086:
20075:
20069:
20064:
20058:
20052:
20046:
20040:
20034:
20028:
20022:
20016:
20010:
20004:
19998:
19992:
19986:
19981:
19975:
19970:
19964:
19952:
19946:
19940:
19931:
19919:
19913:
19907:
19901:
19895:
19889:
19880:
19874:
19869:
19863:
19857:
19851:
19850:
19843:
19837:
19831:
19825:
19819:
19813:
19807:
19801:
19795:
19789:
19783:
19777:
19776:
19765:
19759:
19753:
19747:
19741:
19732:
19726:
19720:
19714:
19698:
19691:
19685:
19682:
19676:
19674:
19669:
19663:
19650:
19644:
19642:
19638:
19633:
19553:Exponential ring
19491:Categorical ring
19466:
19459:
19449:
19447:
19446:
19441:
19411:
19388:
19384:
19376:
19372:
19358:
19354:
19352:
19350:
19349:
19344:
19342:
19322:
19313:
19309:
19291:
19289:
19288:
19283:
19278:
19264:
19263:
19261:
19260:
19257:
19254:
19249:
19246:
19226:
19225:
19209:
19207:
19206:
19201:
19199:
19182:
19181:
19145:
19144:
19128:
19124:
19122:
19121:
19116:
19110:
19109:
19107:
19102:
19097:
19084:
19082:
19081:
19076:
19070:
19069:
19067:
19062:
19057:
19043:
19041:
19040:
19035:
19029:
19028:
19026:
19021:
19016:
19003:
19001:
19000:
18995:
18989:
18988:
18986:
18981:
18976:
18956:
18954:
18953:
18948:
18942:
18941:
18939:
18934:
18929:
18909:
18905:
18897:
18885:
18860:
18858:
18857:
18852:
18809:
18801:
18789:
18781:
18705:
18700:set of operators
18697:
18693:
18689:
18685:
18671:
18667:
18663:
18659:
18655:
18651:
18647:
18643:
18616:
18612:
18608:
18594:
18590:
18582:
18571:
18563:
18559:
18555:
18551:
18543:
18527:
18523:
18504:
18497:
18495:
18494:
18489:
18487:
18429:
18368:
18354:
18344:
18334:
18326:
18322:
18314:
18310:
18306:
18294:
18286:
18271:
18265:
18260:
18254:
18252:
18250:
18249:
18244:
18242:
18131:character theory
18059:
18049:
18044:multilinear form
18041:
18009:
18007:
18006:
18001:
17996:
17982:
17981:
17955:
17953:
17952:
17947:
17939:
17925:
17924:
17914:
17909:
17888:
17874:
17873:
17851:
17798:
17796:
17795:
17790:
17785:
17784:
17783:
17770:
17751:
17750:
17734:
17732:
17730:
17729:
17724:
17722:
17702:
17700:
17699:
17694:
17680:
17679:
17655:
17654:
17644:
17639:
17612:
17611:
17589:
17585:
17571:
17559:Zariski topology
17552:
17548:
17543:product topology
17540:
17526:
17511:
17509:
17508:
17503:
17469:
17464:topological ring
17461:
17454:
17448:
17446:
17444:
17443:
17438:
17436:
17422:
17418:
17414:
17375:
17364:
17360:
17356:
17352:
17348:
17340:
17336:
17325:
17321:
17319:
17318:
17313:
17277:
17220:
17216:
17212:
17210:
17209:
17204:
17175:
17171:
17165:
17163:
17162:
17157:
17122:
17118:
17114:
17103:
17099:
17095:
17093:
17092:
17087:
17048:
17037:
17033:
17029:
17021:
16990:
16980:
16976:
16970:
16964:
16960:
16954:
16947:
16928:Azumaya algebras
16924:
16922:
16921:
16916:
16911:
16907:
16906:
16905:
16887:
16865:
16864:
16848:
16836:
16832:
16828:
16824:
16822:
16821:
16816:
16811:
16810:
16791:
16779:
16767:
16765:
16764:
16759:
16754:
16753:
16734:
16730:
16719:
16717:
16716:
16711:
16709:
16704:
16699:
16669:
16667:
16665:
16663:
16662:
16657:
16655:
16654:
16649:
16629:
16621:
16619:
16618:
16613:
16608:
16579:
16575:
16560:
16552:
16544:
16542:
16541:
16536:
16534:
16530:
16526:
16525:
16480:
16478:
16477:
16472:
16467:
16466:
16451:
16450:
16441:
16440:
16431:
16430:
16414:
16412:
16411:
16406:
16401:
16400:
16391:
16390:
16375:
16374:
16365:
16364:
16345:
16341:
16333:
16329:
16315:
16304:
16300:
16296:
16292:
16288:
16280:
16276:
16272:
16250:
16246:
16233:
16231:
16230:
16225:
16220:
16219:
16196:
16192:
16180:
16169:
16158:
16156:
16155:
16150:
16145:
16144:
16129:
16128:
16127:
16126:
16111:
16106:
16081:
16067:
16061:
16054:
16013:
16005:
15997:
15991:
15987:
15976:
15962:
15958:
15913:Semisimple rings
15901:, introduced by
15823:GCD domains
15785:class inclusions
15764:
15756:
15743:
15736:
15728:
15724:
15717:
15715:
15714:
15709:
15704:
15703:
15676:
15675:
15659:
15657:
15656:
15651:
15646:
15641:
15640:
15639:
15629:
15616:
15588:
15584:
15578:
15574:
15572:
15571:
15566:
15528:
15526:
15525:
15520:
15514:
15513:
15512:
15502:
15489:
15488:
15487:
15477:
15455:
15451:
15445:
15441:
15427:
15423:
15350:
15348:
15347:
15342:
15282:
15278:
15276:
15275:
15270:
15265:
15264:
15245:
15241:
15237:
15231:
15210:
15208:
15207:
15202:
15197:
15194:
15176:
15172:
15164:
15160:
15156:
15154:
15153:
15148:
15091:
15087:
15085:
15083:
15082:
15077:
15072:
15058:
15054:
15050:
15046:
15042:
15038:
15034:
15030:
15023:
15013:
15010:factors through
15009:
15005:
15001:
14997:
14993:
14985:
14967:integral closure
14951:
14943:
14937:
14928:
14926:
14924:
14923:
14918:
14913:
14912:
14907:
14892:
14884:
14876:
14872:
14870:
14868:
14867:
14862:
14860:
14859:
14854:
14839:
14837:
14835:
14834:
14829:
14824:
14823:
14818:
14799:
14797:
14795:
14794:
14789:
14787:
14773:
14771:
14769:
14768:
14763:
14761:
14747:
14745:
14744:
14739:
14737:
14736:
14731:
14722:
14717:
14712:
14711:
14706:
14697:
14689:
14688:
14683:
14674:
14666:
14654:
14652:
14651:
14646:
14638:
14637:
14632:
14623:
14611:
14607:
14603:
14601:
14600:
14595:
14584:
14583:
14567:
14565:
14564:
14559:
14557:
14556:
14546:
14545:
14525:
14524:
14519:
14510:
14498:
14496:
14494:
14493:
14488:
14486:
14472:
14470:
14468:
14467:
14462:
14460:
14446:
14444:
14443:
14438:
14436:
14428:
14410:
14408:
14406:
14405:
14400:
14398:
14384:
14380:
14378:
14376:
14375:
14370:
14365:
14349:
14344:
14342:
14340:
14339:
14334:
14329:
14328:
14323:
14306:
14301:
14297:
14289:
14287:
14285:
14284:
14279:
14277:
14260:
14252:
14248:
14244:
14240:
14236:
14234:
14233:
14228:
14223:
14222:
14214:
14198:
14196:
14195:
14190:
14188:
14187:
14178:
14163:
14159:
14157:
14156:
14151:
14146:
14145:
14136:
14125:
14112:
14111:
14103:
14093:
14089:
14079:
14075:
14071:
14059:
14057:
14056:
14051:
14049:
14045:
14044:
14019:-modules map to
14018:
14014:
14012:
14011:
14006:
14004:
14000:
13999:
13973:
13969:
13965:
13957:
13953:
13949:
13945:
13941:
13921:
13919:
13918:
13913:
13905:
13888:
13867:
13865:
13864:
13859:
13854:
13853:
13840:
13838:
13837:
13832:
13820:
13819:
13806:
13805:
13804:
13787:
13786:
13760:
13756:
13754:
13753:
13748:
13740:
13723:
13702:
13700:
13699:
13694:
13692:
13688:
13687:
13661:
13659:
13658:
13653:
13645:
13641:
13640:
13624:
13613:
13609:
13608:
13586:
13582:
13581:
13565:
13541:
13537:
13534:
13532:
13531:
13526:
13521:
13517:
13516:
13494:
13481:
13477:
13476:
13449:
13447:
13446:
13441:
13436:
13432:
13431:
13405:
13401:
13397:
13395:
13394:
13389:
13387:
13383:
13382:
13366:
13365:
13356:
13355:
13339:
13335:
13329:
13327:
13326:
13321:
13313:
13312:
13303:
13299:
13298:
13276:
13272:
13271:
13242:
13239:with respect to
13238:
13234:
13230:
13224:
13222:
13221:
13216:
13208:
13207:
13188:
13186:
13185:
13180:
13178:
13177:
13176:
13159:
13157:
13156:
13151:
13149:
13148:
13142:
13127:
13123:
13121:
13120:
13115:
13113:
13112:
13099:
13095:
13091:
13089:
13088:
13083:
13078:
13077:
13076:
13066:
13065:
13045:
13043:
13042:
13037:
13035:
13034:
13033:
13017:
13015:
13014:
13009:
13004:
13000:
12999:
12973:
12971:
12970:
12965:
12963:
12962:
12961:
12944:
12942:
12941:
12936:
12931:
12930:
12905:
12901:
12894:
12892:
12891:
12886:
12863:
12843:
12839:
12838:
12813:(to be precise,
12812:
12810:
12809:
12804:
12776:
12775:
12766:
12751:
12749:
12748:
12743:
12741:
12737:
12736:
12710:
12706:
12702:
12698:
12695:with respect to
12694:
12686:
12684:
12683:
12678:
12676:
12672:
12671:
12645:
12643:
12642:
12637:
12632:
12628:
12627:
12601:
12597:
12593:
12591:
12590:
12585:
12580:
12576:
12575:
12549:
12545:
12541:
12539:
12538:
12533:
12531:
12527:
12526:
12494:
12492:
12491:
12486:
12481:
12480:
12455:
12451:
12447:
12418:
12408:
12388:
12377:
12366:
12353:
12351:
12350:
12345:
12342:
12341:
12321:
12319:
12318:
12313:
12311:
12310:
12298:
12281:
12267:
12247:
12218:
12198:
12188:
12168:
12162:
12146:projective limit
12124:
12120:
12114:
12112:
12111:
12106:
12089:
12072:
12061:
12059:
12058:
12053:
12018:
12016:
12015:
12010:
11988:
11987:
11972:
11932:
11922:
11920:
11919:
11914:
11909:
11908:
11907:
11906:
11896:
11887:
11874:
11873:
11868:
11863:
11858:
11838:
11836:
11835:
11830:
11822:
11821:
11803:
11802:
11790:
11789:
11771:
11749:
11748:
11736:
11735:
11706:
11702:
11695:
11685:
11675:
11668:
11666:
11665:
11660:
11658:
11657:
11645:
11628:
11620:filtered colimit
11617:
11613:
11602:is a subring of
11601:
11594:
11579:
11565:
11561:
11547:
11535:
11521:
11503:
11501:
11500:
11495:
11484:
11483:
11468:
11467:
11452:
11451:
11450:
11449:
11434:
11429:
11399:
11398:
11379:
11377:
11376:
11371:
11366:
11365:
11349:
11345:
11338:
11336:
11335:
11330:
11327:
11326:
11325:
11312:
11302:
11297:
11269:
11255:
11251:
11241:
11217:
11204:
11202:
11201:
11196:
11188:
11187:
11150:
11149:
11130:
11125:
11110:
11109:
11090:
11073:
11066:
11062:
11058:
11056:
11055:
11050:
11044:
11039:
11023:
11018:
10996:
10994:
10993:
10988:
10971:
10970:
10955:
10954:
10939:
10938:
10919:
10905:
10897:
10893:
10889:
10885:
10882:. Given a right
10881:
10863:
10860:with entries in
10859:
10855:
10824:
10815:
10813:
10812:
10807:
10799:
10798:
10785:
10784:
10775:
10774:
10764:
10759:
10740:
10738:
10737:
10732:
10693:
10689:
10687:
10686:
10681:
10679:
10675:
10674:
10673:
10655:
10654:
10626:
10619:
10613:
10609:
10605:
10601:
10595:
10588:
10584:
10576:
10574:
10573:
10568:
10566:
10558:
10549:
10545:
10538:
10536:
10535:
10530:
10528:
10520:
10480:
10470:
10466:
10462:
10458:
10454:
10443:
10439:
10437:
10436:
10431:
10429:
10421:
10412:
10410:
10409:
10404:
10387:
10379:
10370:
10368:
10367:
10362:
10339:
10331:
10322:
10318:
10314:
10312:
10311:
10306:
10279:
10275:
10271:
10260:
10233:
10229:
10226:is divisible by
10225:
10202:
10196:
10181:
10177:
10170:
10164:
10158:
10149:
10147:
10146:
10141:
10136:
10132:
10131:
10130:
10118:
10117:
10047:
10045:
10044:
10039:
10037:
10033:
10032:
10031:
10019:
10018:
9991:
9987:
9983:
9979:
9973:
9971:
9969:
9968:
9963:
9934:
9923:
9919:
9905:
9895:
9878:
9876:
9875:
9870:
9818:
9814:
9810:
9808:
9807:
9802:
9781:
9775:
9771:
9765:
9761:
9755:
9747:
9737:
9730:
9728:
9727:
9722:
9720:
9716:
9715:
9692:
9690:
9689:
9684:
9682:
9681:
9663:
9662:
9646:
9644:
9643:
9638:
9636:
9632:
9631:
9630:
9612:
9611:
9587:
9579:
9572:
9570:
9569:
9564:
9562:
9558:
9551:
9550:
9526:
9525:
9510:
9509:
9491:
9490:
9475:
9474:
9456:
9455:
9446:
9445:
9406:
9402:
9375:projective limit
9369:
9365:
9358:
9356:
9355:
9350:
9345:
9344:
9329:
9328:
9323:
9322:
9308:
9298:
9284:
9277:
9275:
9274:
9269:
9264:
9263:
9258:
9257:
9243:
9241:
9240:
9235:
9230:
9229:
9224:
9223:
9213:
9212:
9199:
9198:
9180:
9179:
9157:
9149:
9145:
9143:
9142:
9137:
9135:
9134:
9129:
9128:
9117:
9112:
9107:
9106:
9077:
9076:
9071:
9070:
9063:
9062:
9057:
9056:
9045:
9044:
9039:
9038:
9022:
9021:
9016:
9015:
8995:
8991:
8989:
8988:
8983:
8981:
8980:
8975:
8974:
8960:
8958:
8957:
8952:
8950:
8949:
8944:
8943:
8929:
8927:
8926:
8921:
8919:
8918:
8913:
8912:
8898:
8896:
8895:
8890:
8888:
8887:
8866:
8865:
8849:
8847:
8846:
8841:
8821:
8820:
8802:
8800:
8799:
8794:
8786:
8785:
8780:
8779:
8778:
8755:
8754:
8749:
8748:
8747:
8727:
8726:
8724:
8723:
8718:
8717:
8709:
8704:
8676:
8675:
8674:
8669:
8668:
8661:
8651:
8646:
8628:
8626:
8625:
8624:
8619:
8618:
8609:
8604:
8587:
8568:
8558:
8556:
8555:
8550:
8536:
8535:
8530:
8529:
8519:
8518:
8513:
8512:
8498:
8496:
8495:
8490:
8488:
8487:
8482:
8481:
8465:
8464:
8459:
8458:
8444:
8437:
8435:
8434:
8429:
8427:
8426:
8416:
8394:
8390:
8383:
8379:
8369:
8365:
8355:
8351:
8335:
8331:
8312:
8310:
8309:
8304:
8302:
8295:
8294:
8282:
8281:
8269:
8268:
8256:
8255:
8237:
8236:
8224:
8223:
8205:
8204:
8192:
8191:
8176:
8169:
8168:
8156:
8155:
8143:
8142:
8130:
8129:
8111:
8110:
8098:
8097:
8079:
8078:
8066:
8065:
8050:
8036:
8023:
8019:
7992:
7988:
7984:
7976:
7965:
7961:
7943:
7939:
7935:
7916:additive inverse
7913:
7884:
7864:
7846:
7842:
7823:
7821:
7820:
7815:
7813:
7773:
7756:
7713:
7672:
7659:
7652:
7648:
7642:
7636:
7632:
7626:
7616:
7612:
7608:
7604:
7583:
7579:
7575:
7541:
7538:to the image of
7537:
7527:
7525:
7524:
7519:
7502:
7491:
7483:
7474:
7464:
7447:
7445:
7444:
7439:
7428:
7420:
7405:
7403:
7402:
7397:
7386:
7375:
7367:
7358:
7347:
7329:
7315:
7290:
7280:
7276:
7263:
7261:
7260:
7255:
7253:
7192:
7125:
7111:
7103:
7090:
7082:
7078:
7066:
7060:and a two-sided
7059:
7048:. Given a ring
7025:
7021:
7017:
7009:
7005:
7001:
6994:
6990:
6986:
6982:
6974:
6970:
6948:
6941:-algebra to the
6940:
6933:
6931:
6929:
6928:
6923:
6921:
6913:
6892:
6885:
6883:
6881:
6880:
6875:
6867:
6853:
6846:
6842:
6838:
6817:
6813:
6812:
6801:
6797:
6790:
6782:
6780:
6779:
6774:
6772:
6771:
6728:
6724:
6717:
6715:
6713:
6712:
6707:
6705:
6697:
6692:
6678:
6676:
6674:
6673:
6668:
6666:
6652:
6648:
6644:
6628:inverse function
6625:
6615:
6605:
6603:
6602:
6597:
6595:
6591:
6590:
6575:
6574:
6558:
6505:
6452:
6438:
6434:
6428:
6422:
6418:
6414:
6410:
6402:
6370:
6360:
6350:
6340:
6334:
6328:
6324:
6322:
6321:
6316:
6295:
6293:
6292:
6287:
6269:
6267:
6266:
6261:
6240:
6238:
6237:
6232:
6204:
6200:
6167:
6163:
6159:
6155:
6147:
6141:
6133:
6127:
6121:
6117:
6110:
6106:
6102:
6098:
6092:
6088:
6084:
6073:is said to be a
6072:
6068:
6058:
6050:
6040:
6033:
6031:
6030:
6025:
6014:
6013:
6003:
6002:
5999:
5989:
5988:
5978:
5977:
5974:
5970:
5969:
5966:
5962:
5961:
5952:
5951:
5933:
5932:
5923:
5922:
5903:
5899:
5895:
5891:
5887:
5883:
5873:
5869:
5865:
5861:
5857:
5853:
5849:
5841:
5823:
5815:
5811:
5803:
5799:
5795:
5791:
5787:
5783:
5779:
5775:
5767:
5763:
5755:
5751:
5747:
5737:
5733:
5729:
5725:
5721:
5713:
5702:
5698:
5691:
5681:
5677:
5673:
5669:
5662:
5658:
5654:
5646:
5642:
5634:
5628:
5625:containing
5624:
5620:
5616:
5612:
5608:
5597:
5595:
5593:
5592:
5587:
5585:
5568:
5566:
5564:
5563:
5558:
5553:
5539:
5535:
5533:
5531:
5530:
5525:
5523:
5506:
5504:
5502:
5501:
5496:
5494:
5481:(in both cases,
5480:
5478:
5476:
5475:
5470:
5459:
5438:
5436:
5434:
5433:
5428:
5426:
5408:
5398:
5392:
5388:
5384:
5374:
5370:
5366:
5362:
5353:
5349:
5345:
5328:
5316:
5312:
5290:
5286:
5280:
5276:
5272:
5261:
5254:
5244:
5234:
5219:
5209:
5207:
5206:
5201:
5178:nilpotent matrix
5175:
5168:
5161:
5150:
5143:
5139:
5135:
5131:
5115:
5104:
5091:
5084:
5071:
5064:
5057:
5046:
5017:
5007:
5005:
5004:
4999:
4997:
4996:
4986:
4981:
4963:
4962:
4946:
4942:
4938:
4936:
4935:
4930:
4925:
4924:
4906:
4905:
4886:
4868:
4860:
4856:
4854:
4853:
4848:
4805:
4800:
4750:
4739:
4735:
4733:
4731:
4730:
4725:
4720:
4706:
4702:
4690:
4688:
4686:
4685:
4680:
4669:
4652:
4650:
4648:
4647:
4642:
4640:
4602:
4596:
4592:
4586:
4580:
4572:
4568:
4558:
4550:
4534:
4520:
4516:
4508:
4504:
4498:
4487:
4473:
4464:
4460:
4453:
4449:
4442:
4435:
4431:
4427:
4423:
4397:
4389:
4375:
4371:
4369:
4367:
4366:
4361:
4359:
4358:
4353:
4338:
4334:
4332:
4330:
4329:
4324:
4319:
4318:
4313:
4308:
4286:
4284:
4282:
4281:
4276:
4271:
4257:
4255:
4253:
4252:
4247:
4245:
4222:
4218:
4214:
4210:
4206:
4195:
4193:
4191:
4190:
4185:
4183:
4158:
4150:
4148:
4147:
4142:
4134:
4133:
4115:
4114:
4086:
4078:
4067:
4062:
4001:
3991:. Likewise, the
3990:
3985:
3981:
3953:non-zero-divisor
3938:
3932:
3926:
3922:
3913:
3911:
3910:
3905:
3903:
3889:
3863:
3862:
3840:
3839:
3808:
3807:
3788:
3787:
3756:
3755:
3736:
3735:
3713:
3712:
3693:
3692:
3657:
3656:
3633:
3600:Richard Dedekind
3592:polynomial rings
3576:Richard Dedekind
3550:
3547:with entries in
3546:
3542:
3538:
3531:
3529:
3528:
3523:
3518:
3514:
3464:
3462:
3461:
3456:
3454:
3450:
3400:
3398:
3397:
3392:
3387:
3383:
3336:
3334:
3333:
3328:
3326:
3322:
3275:
3273:
3272:
3267:
3265:
3261:
3220:
3218:
3217:
3212:
3198:
3197:
3174:
3172:
3171:
3166:
3161:
3157:
3127:
3126:
3122:
3121:
3063:
3062:
3043:
3025:
3023:
3022:
3017:
3012:
3004:
2999:
2994:
2986:
2981:
2973:
2961:
2959:
2958:
2953:
2948:
2943:
2935:
2930:
2922:
2910:
2908:
2906:
2905:
2900:
2898:
2890:
2885:
2871:
2869:
2868:
2863:
2861:
2853:
2844:
2840:
2838:
2837:
2832:
2827:
2819:
2810:
2803:
2801:
2799:
2798:
2793:
2788:
2780:
2775:
2761:
2758:when divided by
2757:
2753:
2749:
2747:
2745:
2744:
2739:
2734:
2720:
2718:
2716:
2715:
2710:
2708:
2700:
2695:
2678:
2676:
2675:
2670:
2665:
2657:
2652:
2644:
2639:
2631:
2622:
2620:
2619:
2614:
2612:
2604:
2599:
2591:
2586:
2578:
2569:
2565:
2561:
2559:
2557:
2556:
2551:
2549:
2541:
2536:
2522:
2520:
2519:
2514:
2512:
2504:
2499:
2491:
2480:
2478:
2477:
2472:
2467:
2459:
2454:
2446:
2441:
2433:
2424:
2422:
2421:
2416:
2414:
2406:
2401:
2393:
2388:
2380:
2372:). For example,
2371:
2360:
2350:
2346:
2336:
2332:
2322:
2320:
2318:
2317:
2312:
2310:
2302:
2297:
2283:
2281:
2280:
2275:
2273:
2265:
2260:
2252:
2239:
2237:
2236:
2231:
2229:
2225:
2224:
2216:
2211:
2203:
2198:
2190:
2185:
2177:
2167:
2159:
2154:
2127:
2117:
2113:
2108:binomial formula
2102:
2098:
2087:
2083:
2079:
2075:
2068:
2053:
2043:
2039:
2036:For any element
2011:
2009:
2008:
2003:
1898:
1896:
1894:
1893:
1888:
1883:
1834:
1816:
1792:commutative ring
1779:
1773:
1752:
1751:
1748:
1747:
1744:
1741:
1719:
1713:
1703:
1693:
1689:
1675:
1641:
1637:
1623:
1577:
1573:
1569:
1565:
1554:
1544:
1540:
1534:is associative).
1533:
1529:
1525:
1511:
1477:
1469:
1464:additive inverse
1461:
1454:
1443:
1439:
1432:
1428:
1417:
1413:
1409:
1405:
1395:
1391:
1380:
1376:
1372:
1362:
1338:
1334:
1330:
1316:
1282:
1271:
1243:invariant theory
1235:polynomial rings
1227:Dedekind domains
1200:cohomology rings
1177:
1166:
1155:ring of integers
1122:, the theory of
1103:identity element
1067:
1060:
1053:
842:Commutative ring
771:Rack and quandle
736:
735:
673:that generalize
651:
644:
637:
622:Operator algebra
608:Clifford algebra
544:
542:
541:
536:
531:
530:
518:
497:
495:
494:
489:
487:
486:
481:
460:
458:
457:
452:
450:
449:
444:
422:Ring of integers
416:
413:Integers modulo
364:Euclidean domain
252:
250:
249:
244:
242:
234:
229:
206:
204:
203:
198:
196:
100:Product of rings
86:Fractional ideal
45:
37:
36:
21:
21659:
21658:
21654:
21653:
21652:
21650:
21649:
21648:
21629:
21628:
21627:
21585:Kleiner, Israel
21569:10.2307/2974935
21551:Kleiner, Israel
21480:
21475:
21387:
21385:Primary sources
21382:
21376:
21346:
21326:
21256:
21228:
21197:
21145:10.2307/1969205
21119:
21058:
21011:
20993:
20976:
20971:
20947:
20921:
20898:
20870:
20836:
20802:
20776:
20754:
20732:
20703:
20675:
20621:
20613:. Brooks/Cole.
20602:
20588:Herstein, I. N.
20580:
20561:
20530:Eisenbud, David
20523:
20469:Atiyah, Michael
20451:
20446:
20422:
20403:
20386:
20381:
20376:
20372:
20367:Jacobson (2009)
20365:
20361:
20355:Jacobson (2009)
20353:
20349:
20342:
20338:
20332:Milne & CFT
20330:
20326:
20314:
20310:
20302:
20298:
20287:
20283:
20275:
20271:
20263:
20259:
20251:
20247:
20241:Eisenbud (1995)
20239:
20235:
20227:
20223:
20215:
20211:
20203:
20196:
20190:Bourbaki (1964)
20188:
20184:
20178:Jacobson (2009)
20176:
20172:
20168:, Theorem 4.5.1
20164:
20160:
20154:Bourbaki (1989)
20152:
20148:
20142:Jacobson (2009)
20140:
20136:
20128:
20124:
20116:
20112:
20105:
20101:
20094:
20090:
20077:
20076:
20072:
20065:
20061:
20053:
20049:
20041:
20037:
20029:
20025:
20019:Eisenbud (1995)
20017:
20013:
20007:Bourbaki (1989)
20005:
20001:
19993:
19989:
19982:
19978:
19971:
19967:
19963:
19955:Fraenkel (1915)
19953:
19949:
19941:
19934:
19930:
19922:Fraenkel (1915)
19920:
19916:
19910:Jacobson (2009)
19908:
19904:
19898:Fraenkel (1915)
19896:
19892:
19881:
19877:
19870:
19866:
19858:
19854:
19845:
19844:
19840:
19832:
19828:
19820:
19816:
19808:
19804:
19796:
19792:
19784:
19780:
19767:
19766:
19762:
19754:
19750:
19742:
19735:
19727:
19723:
19717:Bourbaki (1989)
19715:
19711:
19707:
19702:
19701:
19692:
19688:
19683:
19679:
19672:
19670:
19666:
19660:next subsection
19651:
19647:
19640:
19636:
19634:
19630:
19625:
19620:
19602:Ring of periods
19530:
19481:
19464:
19462:sphere spectrum
19451:
19450:and a unit map
19417:
19414:
19413:
19409:
19395:
19386:
19382:
19374:
19366:
19360:
19356:
19338:
19330:
19327:
19326:
19324:
19321:
19315:
19311:
19307:
19297:
19265:
19256:
19255:
19250:
19248:
19247:
19230:
19221:
19217:
19215:
19212:
19211:
19186:
19177:
19173:
19140:
19136:
19134:
19131:
19130:
19126:
19103:
19098:
19096:
19095:
19090:
19087:
19086:
19063:
19058:
19056:
19055:
19049:
19046:
19045:
19022:
19017:
19015:
19014:
19009:
19006:
19005:
18982:
18977:
18975:
18974:
18962:
18959:
18958:
18935:
18930:
18928:
18927:
18915:
18912:
18911:
18907:
18903:
18895:
18892:terminal object
18883:
18880:
18875:
18822:
18819:
18818:
18807:
18791:
18783:
18775:
18764:
18748:
18736:
18728:
18703:
18695:
18691:
18687:
18679:
18673:
18669:
18665:
18661:
18657:
18653:
18649:
18645:
18618:
18614:
18610:
18602:
18596:
18592:
18588:
18573:
18565:
18561:
18557:
18553:
18545:
18533:
18525:
18524:is addition in
18506:
18502:
18485:
18484:
18427:
18426:
18365:
18363:
18360:
18359:
18346:
18336:
18328:
18324:
18316:
18312:
18308:
18300:
18288:
18280:
18269:
18263:
18258:
18238:
18236:
18233:
18232:
18230:
18209:
18203:
18179:
18161:. The study of
18159:coordinate ring
18155:valuation rings
18143:
18114:
18085:
18079:and much more.
18051:
18047:
18039:
18036:natural product
17992:
17977:
17973:
17971:
17968:
17967:
17965:homology groups
17935:
17920:
17916:
17910:
17899:
17884:
17869:
17865:
17863:
17860:
17859:
17854:cohomology ring
17849:
17843:
17835:associated ring
17827:
17779:
17766:
17765:
17764:
17746:
17742:
17740:
17737:
17736:
17718:
17716:
17713:
17712:
17710:
17669:
17665:
17650:
17646:
17640:
17635:
17607:
17603:
17601:
17598:
17597:
17592:exterior powers
17587:
17573:
17569:
17550:
17546:
17532:
17513:
17479:
17476:
17475:
17467:
17459:
17450:
17432:
17430:
17427:
17426:
17424:
17420:
17416:
17412:
17393:
17388:
17366:
17362:
17358:
17354:
17350:
17346:
17338:
17327:
17323:
17267:
17234:
17231:
17230:
17218:
17214:
17181:
17178:
17177:
17173:
17169:
17131:
17128:
17127:
17120:
17116:
17105:
17101:
17097:
17064:
17061:
17060:
17039:
17035:
17031:
17027:
16992:
16982:
16978:
16972:
16966:
16962:
16956:
16952:
16945:
16942:
16936:
16901:
16897:
16883:
16870:
16866:
16860:
16856:
16854:
16851:
16850:
16838:
16834:
16830:
16826:
16806:
16802:
16797:
16794:
16793:
16781:
16769:
16749:
16745:
16740:
16737:
16736:
16732:
16728:
16705:
16700:
16695:
16675:
16672:
16671:
16650:
16645:
16644:
16642:
16639:
16638:
16636:
16635:
16627:
16626:). Finally, if
16604:
16593:
16590:
16589:
16577:
16569:
16554:
16550:
16521:
16517:
16513:
16509:
16486:
16483:
16482:
16459:
16455:
16446:
16442:
16436:
16432:
16426:
16422:
16420:
16417:
16416:
16396:
16392:
16386:
16382:
16370:
16366:
16360:
16356:
16351:
16348:
16347:
16343:
16339:
16334:are said to be
16331:
16327:
16314:
16306:
16302:
16298:
16294:
16290:
16286:
16278:
16274:
16270:
16267:
16261:
16248:
16238:
16236:field extension
16215:
16211:
16206:
16203:
16202:
16194:
16190:
16179:
16171:
16168:
16160:
16140:
16136:
16122:
16118:
16117:
16113:
16107:
16096:
16090:
16087:
16086:
16079:
16065:
16059:
16052:
16046:
16024:are semisimple.
16011:
16003:
15993:
15989:
15985:
15970:
15964:
15960:
15956:
15941:
15932:semisimple ring
15921:
15915:
15862:
15762:
15754:
15745:
15742:
15738:
15734:
15726:
15723:
15719:
15699:
15695:
15671:
15667:
15665:
15662:
15661:
15635:
15631:
15630:
15625:
15617:
15612:
15598:
15595:
15594:
15586:
15580:
15576:
15536:
15533:
15532:
15508:
15504:
15503:
15498:
15483:
15479:
15478:
15473:
15461:
15458:
15457:
15453:
15447:
15443:
15429:
15425:
15421:
15387:integral domain
15371:
15366:
15361:
15359:Change of rings
15288:
15285:
15284:
15280:
15260:
15256:
15251:
15248:
15247:
15243:
15239:
15233:
15227:
15193:
15185:
15182:
15181:
15174:
15170:
15162:
15158:
15097:
15094:
15093:
15089:
15068:
15066:
15063:
15062:
15060:
15056:
15052:
15048:
15044:
15040:
15036:
15035:is a subset of
15032:
15028:
15015:
15011:
15007:
15003:
14999:
14995:
14991:
14983:
14980:
14945:
14939:
14932:
14908:
14903:
14902:
14900:
14897:
14896:
14894:
14890:
14880:
14878:
14874:
14855:
14850:
14849:
14847:
14844:
14843:
14841:
14819:
14814:
14813:
14811:
14808:
14807:
14805:
14783:
14781:
14778:
14777:
14775:
14757:
14755:
14752:
14751:
14749:
14732:
14727:
14726:
14718:
14713:
14707:
14702:
14701:
14693:
14684:
14679:
14678:
14670:
14662:
14660:
14657:
14656:
14633:
14628:
14627:
14619:
14617:
14614:
14613:
14609:
14605:
14579:
14575:
14573:
14570:
14569:
14541:
14537:
14533:
14529:
14520:
14515:
14514:
14506:
14504:
14501:
14500:
14482:
14480:
14477:
14476:
14474:
14456:
14454:
14451:
14450:
14448:
14432:
14424:
14416:
14413:
14412:
14394:
14392:
14389:
14388:
14386:
14382:
14361:
14359:
14356:
14355:
14353:
14347:
14324:
14319:
14318:
14316:
14313:
14312:
14310:
14309:and is denoted
14304:
14299:
14291:
14273:
14271:
14268:
14267:
14265:
14258:
14250:
14246:
14242:
14238:
14213:
14212:
14204:
14201:
14200:
14183:
14179:
14174:
14169:
14166:
14165:
14161:
14141:
14137:
14132:
14116:
14102:
14101:
14099:
14096:
14095:
14091:
14087:
14077:
14073:
14069:
14066:
14037:
14033:
14029:
14024:
14021:
14020:
14016:
13992:
13988:
13984:
13979:
13976:
13975:
13971:
13967:
13963:
13955:
13951:
13947:
13943:
13939:
13932:category theory
13898:
13881:
13873:
13870:
13869:
13849:
13848:
13846:
13843:
13842:
13815:
13814:
13800:
13799:
13795:
13782:
13781:
13769:
13766:
13765:
13764:Conversely, if
13758:
13733:
13716:
13708:
13705:
13704:
13680:
13676:
13672:
13667:
13664:
13663:
13633:
13629:
13625:
13617:
13601:
13597:
13593:
13574:
13570:
13566:
13558:
13550:
13547:
13546:
13539:
13535:
13509:
13505:
13501:
13485:
13469:
13465:
13461:
13456:
13453:
13452:
13424:
13420:
13416:
13411:
13408:
13407:
13403:
13399:
13375:
13371:
13367:
13361:
13360:
13351:
13350:
13348:
13345:
13344:
13337:
13333:
13308:
13304:
13291:
13287:
13283:
13264:
13260:
13256:
13251:
13248:
13247:
13245:change of rings
13240:
13236:
13232:
13228:
13203:
13202:
13194:
13191:
13190:
13172:
13171:
13167:
13165:
13162:
13161:
13144:
13143:
13138:
13133:
13130:
13129:
13125:
13108:
13107:
13105:
13102:
13101:
13097:
13093:
13072:
13071:
13067:
13061:
13060:
13058:
13055:
13054:
13029:
13028:
13024:
13022:
13019:
13018:
12992:
12988:
12984:
12979:
12976:
12975:
12957:
12956:
12952:
12950:
12947:
12946:
12926:
12925:
12911:
12908:
12907:
12906:. In that case
12903:
12899:
12859:
12831:
12827:
12823:
12818:
12815:
12814:
12771:
12767:
12762:
12757:
12754:
12753:
12729:
12725:
12721:
12716:
12713:
12712:
12708:
12704:
12700:
12696:
12692:
12664:
12660:
12656:
12651:
12648:
12647:
12620:
12616:
12612:
12607:
12604:
12603:
12599:
12598:that "inverts"
12595:
12568:
12564:
12560:
12555:
12552:
12551:
12547:
12543:
12542:that "inverts"
12519:
12515:
12511:
12500:
12497:
12496:
12473:
12469:
12461:
12458:
12457:
12453:
12449:
12445:
12434:
12410:
12407:
12398:
12390:
12387:
12379:
12376:
12368:
12364:
12355:
12337:
12333:
12327:
12324:
12323:
12306:
12302:
12289:
12287:
12284:
12283:
12269:
12266:
12257:
12249:
12246:
12237:
12228:
12220:
12217:
12208:
12200:
12190:
12187:
12178:
12170:
12164:
12161:
12153:
12122:
12116:
12080:
12078:
12075:
12074:
12070:
12027:
12024:
12023:
11980:
11976:
11963:
11938:
11935:
11934:
11930:
11902:
11898:
11897:
11892:
11891:
11878:
11869:
11859:
11857:
11856:
11854:
11851:
11850:
11817:
11813:
11798:
11794:
11785:
11781:
11762:
11744:
11740:
11731:
11727:
11719:
11716:
11715:
11704:
11701:
11697:
11687:
11686:if and only if
11677:
11674:
11670:
11653:
11649:
11636:
11634:
11631:
11630:
11627:
11623:
11615:
11612:
11603:
11600:
11596:
11593:
11589:
11586:
11573:
11567:
11563:
11555:
11549:
11545:
11529:
11523:
11519:
11513:semisimple ring
11479:
11475:
11463:
11459:
11445:
11441:
11440:
11436:
11430:
11419:
11394:
11390:
11388:
11385:
11384:
11361:
11357:
11355:
11352:
11351:
11347:
11344:
11340:
11321:
11317:
11313:
11308:
11298:
11287:
11275:
11272:
11271:
11263:
11257:
11253:
11249:
11235:
11225:
11219:
11209:
11180:
11176:
11145:
11141:
11126:
11121:
11105:
11101:
11099:
11096:
11095:
11084:
11075:
11072:
11068:
11064:
11060:
11040:
11035:
11019:
11014:
11002:
10999:
10998:
10966:
10962:
10950:
10946:
10934:
10930:
10928:
10925:
10924:
10913:
10907:
10903:
10895:
10891:
10887:
10883:
10875:
10869:
10861:
10857:
10853:
10850:
10842:Main articles:
10840:
10820:
10794:
10790:
10780:
10776:
10770:
10766:
10760:
10755:
10749:
10746:
10745:
10709:
10706:
10705:
10691:
10669:
10665:
10650:
10646:
10645:
10641:
10636:
10633:
10632:
10624:
10615:
10611:
10607:
10603:
10597:
10593:
10586:
10582:
10557:
10555:
10552:
10551:
10547:
10543:
10519:
10489:
10486:
10485:
10472:
10468:
10464:
10460:
10456:
10452:
10441:
10420:
10418:
10415:
10414:
10378:
10376:
10373:
10372:
10330:
10328:
10325:
10324:
10320:
10316:
10315:and an element
10288:
10285:
10284:
10277:
10273:
10262:
10235:
10231:
10227:
10204:
10198:
10183:
10179:
10175:
10166:
10160:
10154:
10126:
10122:
10113:
10109:
10108:
10104:
10056:
10053:
10052:
10027:
10023:
10014:
10010:
10009:
10005:
10000:
9997:
9996:
9989:
9985:
9981:
9975:
9942:
9939:
9938:
9936:
9925:
9921:
9907:
9897:
9887:
9827:
9824:
9823:
9816:
9812:
9790:
9787:
9786:
9777:
9773:
9767:
9763:
9757:
9753:
9743:
9740:integral domain
9735:
9711:
9707:
9703:
9698:
9695:
9694:
9677:
9673:
9658:
9654:
9652:
9649:
9648:
9626:
9622:
9607:
9603:
9602:
9598:
9593:
9590:
9589:
9585:
9582:polynomial ring
9577:
9546:
9542:
9521:
9517:
9505:
9501:
9480:
9476:
9464:
9460:
9451:
9447:
9441:
9437:
9436:
9432:
9415:
9412:
9411:
9404:
9400:
9399:Given a symbol
9397:
9395:Polynomial ring
9391:
9389:Polynomial ring
9367:
9364:
9360:
9340:
9336:
9324:
9318:
9317:
9316:
9314:
9311:
9310:
9300:
9295:
9291:
9286:
9283:
9279:
9259:
9253:
9252:
9251:
9249:
9246:
9245:
9225:
9219:
9218:
9217:
9208:
9204:
9194:
9190:
9175:
9171:
9163:
9160:
9159:
9155:
9147:
9130:
9124:
9123:
9122:
9113:
9108:
9102:
9101:
9072:
9066:
9065:
9064:
9058:
9052:
9051:
9050:
9040:
9034:
9033:
9032:
9017:
9011:
9010:
9009:
9001:
8998:
8997:
8993:
8976:
8970:
8969:
8968:
8966:
8963:
8962:
8945:
8939:
8938:
8937:
8935:
8932:
8931:
8930:(in particular
8914:
8908:
8907:
8906:
8904:
8901:
8900:
8883:
8879:
8861:
8857:
8855:
8852:
8851:
8816:
8812:
8810:
8807:
8806:
8781:
8774:
8773:
8769:
8768:
8750:
8743:
8742:
8738:
8737:
8719:
8713:
8712:
8711:
8705:
8694:
8688:
8684:
8670:
8664:
8663:
8662:
8657:
8653:
8647:
8636:
8620:
8614:
8613:
8612:
8611:
8605:
8594:
8588:
8583:
8578:
8575:
8574:
8560:
8531:
8525:
8524:
8523:
8514:
8508:
8507:
8506:
8504:
8501:
8500:
8483:
8477:
8476:
8475:
8460:
8454:
8453:
8452:
8450:
8447:
8446:
8442:
8422:
8418:
8406:
8400:
8397:
8396:
8392:
8389:
8385:
8381:
8377:
8367:
8357:
8353:
8350:
8343:
8337:
8333:
8330:
8323:
8317:
8300:
8299:
8290:
8286:
8277:
8273:
8264:
8260:
8251:
8247:
8232:
8228:
8219:
8215:
8200:
8196:
8187:
8183:
8174:
8173:
8164:
8160:
8151:
8147:
8138:
8134:
8125:
8121:
8106:
8102:
8093:
8089:
8074:
8070:
8061:
8057:
8047:
8045:
8042:
8041:
8028:
8021:
8017:
8014:
8008:
8003:
7990:
7986:
7982:
7967:
7963:
7945:
7941:
7937:
7923:
7904:
7866:
7848:
7844:
7840:
7811:
7810:
7771:
7770:
7754:
7753:
7711:
7710:
7669:
7667:
7664:
7663:
7657:
7650:
7644:
7638:
7634:
7628:
7622:
7614:
7610:
7606:
7592:
7581:
7577:
7573:
7554:
7548:
7539:
7529:
7498:
7482:
7480:
7477:
7476:
7466:
7452:
7419:
7411:
7408:
7407:
7382:
7366:
7364:
7361:
7360:
7349:
7335:
7317:
7299:
7282:
7278:
7268:
7251:
7250:
7190:
7189:
7122:
7120:
7117:
7116:
7109:
7095:
7084:
7083:as subgroup of
7080:
7068:
7064:
7049:
7038:
7032:
7023:
7019:
7015:
7007:
7003:
6999:
6992:
6988:
6984:
6980:
6972:
6958:
6946:
6938:
6917:
6909:
6907:
6904:
6903:
6901:
6887:
6863:
6861:
6858:
6857:
6855:
6851:
6844:
6840:
6830:
6815:
6808:
6803:
6799:
6795:
6788:
6764:
6760:
6734:
6731:
6730:
6726:
6722:
6701:
6693:
6688:
6686:
6683:
6682:
6680:
6662:
6660:
6657:
6656:
6654:
6650:
6646:
6642:
6623:
6613:
6593:
6592:
6586:
6582:
6570:
6566:
6556:
6555:
6503:
6502:
6449:
6447:
6444:
6443:
6436:
6430:
6424:
6420:
6416:
6412:
6404:
6392:
6383:
6377:
6362:
6352:
6351:implies either
6342:
6336:
6330:
6326:
6301:
6298:
6297:
6275:
6272:
6271:
6270:implies either
6246:
6243:
6242:
6214:
6211:
6210:
6202:
6198:
6184:Noetherian ring
6165:
6161:
6157:
6149:
6143:
6139:
6129:
6123:
6119:
6115:
6108:
6104:
6100:
6094:
6090:
6089:is a subset of
6086:
6082:
6075:two-sided ideal
6070:
6060:
6056:
6051:. Similarly, a
6042:
6038:
6009:
6005:
5998:
5997:
5984:
5980:
5973:
5972:
5965:
5964:
5957:
5953:
5947:
5943:
5928:
5924:
5918:
5914:
5912:
5909:
5908:
5901:
5897:
5893:
5889:
5885:
5875:
5874:, the elements
5871:
5867:
5863:
5859:
5855:
5851:
5847:
5839:
5836:
5830:
5821:
5813:
5809:
5801:
5797:
5793:
5789:
5785:
5781:
5777:
5773:
5765:
5757:
5753:
5749:
5739:
5735:
5731:
5727:
5723:
5715:
5711:
5700:
5693:
5689:
5679:
5675:
5671:
5664:
5660:
5656:
5652:
5644:
5640:
5632:
5626:
5622:
5618:
5614:
5610:
5606:
5581:
5576:
5573:
5572:
5570:
5569:one could call
5549:
5547:
5544:
5543:
5541:
5537:
5519:
5514:
5511:
5510:
5508:
5490:
5488:
5485:
5484:
5482:
5455:
5453:
5450:
5449:
5447:
5422:
5420:
5417:
5416:
5414:
5400:
5396:
5390:
5386:
5376:
5372:
5371:, the elements
5368:
5364:
5357:
5351:
5347:
5333:
5326:
5314:
5310:
5307:
5301:
5288:
5282:
5278:
5274:
5263:
5256:
5250:
5240:
5232:
5211:
5195:
5192:
5191:
5170:
5163:
5159:
5145:
5141:
5137:
5133:
5129:
5122:
5106:
5093:
5086:
5073:
5066:
5062:
5048:
5045:
5037:
5027:
5019:
5015:
5009:
4992:
4988:
4982:
4971:
4958:
4954:
4952:
4949:
4948:
4944:
4940:
4920:
4916:
4901:
4897:
4892:
4889:
4888:
4884:
4881:
4876:
4866:
4858:
4801:
4793:
4760:
4757:
4756:
4748:
4737:
4716:
4714:
4711:
4710:
4708:
4704:
4700:
4665:
4660:
4657:
4656:
4654:
4636:
4634:
4631:
4630:
4628:
4626:natural numbers
4621:
4612:Banach algebras
4598:
4594:
4588:
4582:
4578:
4570:
4564:
4559:is a ring, the
4556:
4546:
4528:
4522:
4518:
4514:
4506:
4500:
4492:
4483:
4469:
4462:
4455:
4451:
4444:
4440:
4433:
4429:
4425:
4421:
4417:
4395:
4387:
4373:
4354:
4349:
4348:
4346:
4343:
4342:
4340:
4339:-adic integers
4336:
4309:
4307:
4306:
4304:
4301:
4300:
4298:
4267:
4265:
4262:
4261:
4259:
4241:
4239:
4236:
4235:
4233:
4223:is commutative.
4220:
4216:
4212:
4208:
4204:
4179:
4177:
4174:
4173:
4171:
4162:The set of all
4156:
4129:
4125:
4110:
4106:
4095:
4092:
4091:
4084:
4074:
4065:
4060:
4043:
4038:
4032:
3999:
3988:
3983:
3979:
3973:
3945:
3934:
3928:
3924:
3918:
3901:
3900:
3892:
3883:
3882:
3858:
3854:
3841:
3835:
3831:
3828:
3827:
3803:
3799:
3789:
3783:
3779:
3776:
3775:
3751:
3747:
3737:
3731:
3727:
3724:
3723:
3708:
3704:
3694:
3688:
3684:
3681:
3680:
3658:
3652:
3648:
3644:
3642:
3639:
3638:
3624:
3612:
3588:
3569:
3563:
3548:
3544:
3540:
3536:
3511:
3510:
3505:
3499:
3498:
3493:
3485:
3481:
3470:
3467:
3466:
3447:
3446:
3441:
3435:
3434:
3429:
3421:
3417:
3406:
3403:
3402:
3380:
3379:
3374:
3368:
3367:
3362:
3354:
3350:
3342:
3339:
3338:
3319:
3318:
3313:
3307:
3306:
3301:
3293:
3289:
3281:
3278:
3277:
3258:
3257:
3252:
3246:
3245:
3240:
3232:
3228:
3226:
3223:
3222:
3193:
3189:
3187:
3184:
3183:
3116:
3115:
3110:
3104:
3103:
3098:
3088:
3087:
3084:
3083:
3079:
3058:
3054:
3052:
3049:
3048:
3041:
3035:square matrices
3031:
3003:
2987:
2985:
2972:
2967:
2964:
2963:
2936:
2934:
2921:
2916:
2913:
2912:
2894:
2886:
2881:
2879:
2876:
2875:
2873:
2852:
2850:
2847:
2846:
2842:
2818:
2816:
2813:
2812:
2805:
2784:
2776:
2771:
2769:
2766:
2765:
2763:
2759:
2755:
2751:
2730:
2728:
2725:
2724:
2722:
2704:
2696:
2691:
2689:
2686:
2685:
2683:
2656:
2643:
2630:
2628:
2625:
2624:
2603:
2590:
2577:
2575:
2572:
2571:
2570:. For example,
2567:
2563:
2545:
2537:
2532:
2530:
2527:
2526:
2524:
2503:
2490:
2488:
2485:
2484:
2458:
2445:
2432:
2430:
2427:
2426:
2405:
2392:
2379:
2377:
2374:
2373:
2362:
2352:
2348:
2338:
2334:
2324:
2306:
2298:
2293:
2291:
2288:
2287:
2285:
2264:
2251:
2249:
2246:
2245:
2215:
2202:
2189:
2176:
2175:
2171:
2163:
2155:
2150:
2148:
2145:
2144:
2141:
2135:
2119:
2115:
2111:
2100:
2096:
2085:
2081:
2077:
2073:
2059:
2045:
2041:
2037:
2021:
2019:Some properties
1910:
1907:
1906:
1879:
1877:
1874:
1873:
1871:
1848:
1818:
1814:
1775:
1771:
1738:
1734:
1726:
1715:
1705:
1701:
1691:
1677:
1645:
1639:
1625:
1594:
1575:
1571:
1567:
1556:
1546:
1542:
1538:
1531:
1527:
1513:
1485:
1475:
1467:
1456:
1445:
1441:
1434:
1430:
1426:
1415:
1411:
1407:
1397:
1393:
1389:
1378:
1374:
1364:
1346:
1336:
1332:
1318:
1290:
1280:
1269:
1259:
1172:
1169:square matrices
1158:
1147:coordinate ring
1071:
1042:
1041:
1040:
1011:Non-associative
993:
982:
981:
971:
951:
940:
939:
928:Map of lattices
924:
920:Boolean algebra
915:Heyting algebra
889:
878:
877:
871:
852:Integral domain
816:
805:
804:
798:
752:
723:square matrices
715:complex numbers
655:
626:
625:
558:
548:
547:
526:
522:
514:
512:
509:
508:
482:
477:
476:
474:
471:
470:
445:
440:
439:
437:
434:
433:
414:
384:Polynomial ring
334:Integral domain
323:
313:
312:
238:
230:
225:
217:
214:
213:
192:
190:
187:
186:
172:Involutive ring
57:
46:
40:
35:
28:
23:
22:
15:
12:
11:
5:
21657:
21647:
21646:
21641:
21626:
21625:
21613:
21581:
21563:(5): 417β424.
21547:
21536:
21518:
21502:
21497:
21481:
21479:
21476:
21474:
21473:
21455:(1β2): 24β66.
21437:
21425:Hilbert, David
21421:
21388:
21386:
21383:
21381:
21380:
21374:
21362:Samuel, Pierre
21358:Zariski, Oscar
21354:
21344:
21329:
21324:
21309:
21297:
21260:
21254:
21239:
21226:
21210:
21201:
21195:
21176:
21164:
21123:
21117:
21097:
21088:
21061:
21056:
21036:
21027:
21025:(61): 222β227.
21014:
21009:
20996:
20991:
20977:
20975:
20972:
20970:
20969:
20960:
20951:
20945:
20937:Modern Algebra
20932:
20919:
20901:
20896:
20883:
20874:
20868:
20851:
20849:. AMS Chelsea.
20839:
20834:
20821:
20800:
20780:
20774:
20762:Lam, Tsit Yuen
20758:
20752:
20740:Lam, Tsit Yuen
20736:
20730:
20718:Lam, Tsit Yuen
20714:
20701:
20679:
20673:
20657:
20641:
20625:
20619:
20606:
20600:
20584:
20578:
20565:
20559:
20546:
20526:
20521:
20505:
20493:
20481:
20465:
20457:Artin, Michael
20452:
20450:
20447:
20445:
20444:
20438:Serre (2006),
20435:
20425:
20420:
20407:
20401:
20387:
20385:
20382:
20380:
20379:
20370:
20359:
20347:
20336:
20324:
20308:
20296:
20281:
20269:
20257:
20245:
20243:, Exercise 2.2
20233:
20221:
20209:
20194:
20182:
20170:
20158:
20146:
20134:
20122:
20110:
20099:
20088:
20070:
20067:Garling (2022)
20059:
20047:
20035:
20031:Gallian (2006)
20023:
20011:
19999:
19987:
19976:
19965:
19961:
19947:
19943:Noether (1921)
19932:
19928:
19914:
19902:
19890:
19875:
19872:Hilbert (1897)
19864:
19860:Kleiner (1998)
19852:
19838:
19826:
19814:
19802:
19790:
19778:
19760:
19748:
19733:
19721:
19708:
19706:
19703:
19700:
19699:
19686:
19677:
19664:
19645:
19627:
19626:
19624:
19621:
19619:
19618:
19612:Valuation ring
19609:
19604:
19599:
19594:
19589:
19584:
19579:
19577:artinian rings
19570:
19565:
19560:
19555:
19550:
19545:
19540:
19534:
19529:
19528:
19523:
19518:
19513:
19508:
19503:
19498:
19493:
19488:
19482:
19480:
19477:
19439:
19436:
19433:
19430:
19427:
19424:
19421:
19394:
19391:
19362:
19341:
19337:
19334:
19317:
19296:
19293:
19281:
19277:
19274:
19271:
19268:
19253:
19245:
19242:
19239:
19236:
19233:
19229:
19224:
19220:
19198:
19195:
19192:
19189:
19185:
19180:
19176:
19172:
19169:
19166:
19163:
19160:
19157:
19154:
19151:
19148:
19143:
19139:
19114:
19106:
19101:
19094:
19074:
19066:
19061:
19053:
19033:
19025:
19020:
19013:
18993:
18985:
18980:
18972:
18969:
18966:
18946:
18938:
18933:
18925:
18922:
18919:
18879:
18876:
18874:
18871:
18870:
18869:
18862:
18850:
18847:
18844:
18841:
18838:
18835:
18832:
18829:
18826:
18763:
18760:
18747:
18744:
18735:
18732:
18727:
18726:Generalization
18724:
18675:
18598:
18499:
18498:
18483:
18480:
18477:
18474:
18471:
18468:
18465:
18462:
18459:
18456:
18453:
18450:
18447:
18444:
18441:
18438:
18435:
18432:
18430:
18428:
18425:
18422:
18419:
18416:
18413:
18410:
18407:
18404:
18401:
18398:
18395:
18392:
18389:
18386:
18383:
18380:
18377:
18374:
18371:
18369:
18367:
18241:
18202:
18199:
18178:
18175:
18151:function field
18142:
18139:
18113:
18110:
18084:
18081:
17999:
17995:
17991:
17988:
17985:
17980:
17976:
17957:
17956:
17945:
17942:
17938:
17934:
17931:
17928:
17923:
17919:
17913:
17908:
17905:
17902:
17898:
17894:
17891:
17887:
17883:
17880:
17877:
17872:
17868:
17842:
17839:
17826:
17823:
17822:
17821:
17818:total ordering
17809:
17808:
17788:
17782:
17777:
17774:
17769:
17763:
17760:
17757:
17754:
17749:
17745:
17721:
17706:
17705:
17704:
17703:
17692:
17689:
17686:
17683:
17678:
17675:
17672:
17668:
17664:
17661:
17658:
17653:
17649:
17643:
17638:
17634:
17630:
17627:
17624:
17621:
17618:
17615:
17610:
17606:
17586:that are like
17562:
17501:
17498:
17495:
17492:
17489:
17486:
17483:
17456:
17449:has dimension
17435:
17392:
17389:
17386:uniserial ring
17378:
17377:
17311:
17308:
17305:
17302:
17299:
17296:
17293:
17290:
17287:
17284:
17281:
17276:
17273:
17270:
17266:
17262:
17259:
17256:
17253:
17250:
17247:
17244:
17241:
17238:
17202:
17198:
17195:
17192:
17188:
17185:
17166:
17155:
17152:
17148:
17145:
17142:
17138:
17135:
17085:
17081:
17078:
17075:
17071:
17068:
17024:valuation ring
16948:is a field, a
16940:Valuation ring
16938:Main article:
16935:
16934:Valuation ring
16932:
16914:
16910:
16904:
16900:
16896:
16893:
16890:
16886:
16882:
16879:
16876:
16873:
16869:
16863:
16859:
16814:
16809:
16805:
16801:
16757:
16752:
16748:
16744:
16708:
16703:
16698:
16694:
16691:
16688:
16685:
16682:
16679:
16653:
16648:
16634:(for example,
16611:
16607:
16603:
16600:
16597:
16586:Tsen's theorem
16576:is trivial if
16533:
16529:
16524:
16520:
16516:
16512:
16508:
16505:
16502:
16499:
16496:
16493:
16490:
16470:
16465:
16462:
16458:
16454:
16449:
16445:
16439:
16435:
16429:
16425:
16404:
16399:
16395:
16389:
16385:
16381:
16378:
16373:
16369:
16363:
16359:
16355:
16310:
16263:Main article:
16260:
16257:
16223:
16218:
16214:
16210:
16197:is said to be
16187:
16186:
16175:
16164:
16148:
16143:
16139:
16135:
16132:
16125:
16121:
16116:
16110:
16105:
16102:
16099:
16095:
16084:direct product
16077:
16063:
16062:is semisimple.
16045:
16042:
16026:
16025:
16019:
16000:characteristic
15982:
15966:
15953:
15940:
15937:
15917:Main article:
15914:
15911:
15899:cyclic algebra
15861:
15858:
15857:
15856:
15752:
15740:
15721:
15707:
15702:
15698:
15694:
15691:
15688:
15685:
15682:
15679:
15674:
15670:
15649:
15645:
15638:
15634:
15628:
15624:
15620:
15615:
15611:
15608:
15605:
15602:
15591:cyclic modules
15564:
15561:
15558:
15555:
15552:
15549:
15546:
15543:
15540:
15518:
15511:
15507:
15501:
15497:
15493:
15486:
15482:
15476:
15472:
15468:
15465:
15446:. Then, since
15395:prime elements
15370:
15367:
15365:
15362:
15340:
15337:
15334:
15331:
15328:
15325:
15322:
15319:
15316:
15313:
15310:
15307:
15304:
15301:
15298:
15295:
15292:
15268:
15263:
15259:
15255:
15200:
15192:
15189:
15146:
15143:
15140:
15137:
15134:
15131:
15128:
15125:
15122:
15119:
15116:
15113:
15110:
15107:
15104:
15101:
15075:
15071:
15051:and relations
14979:
14976:
14972:excellent ring
14970:definition of
14955:Hensel's lemma
14916:
14911:
14906:
14886:
14858:
14853:
14827:
14822:
14817:
14804:is denoted by
14786:
14760:
14735:
14730:
14725:
14721:
14716:
14710:
14705:
14700:
14696:
14692:
14687:
14682:
14677:
14673:
14669:
14665:
14644:
14641:
14636:
14631:
14626:
14622:
14593:
14590:
14587:
14582:
14578:
14555:
14552:
14549:
14544:
14540:
14536:
14532:
14528:
14523:
14518:
14513:
14509:
14485:
14459:
14435:
14431:
14427:
14423:
14420:
14397:
14368:
14364:
14332:
14327:
14322:
14307:-adic integers
14276:
14226:
14220:
14217:
14211:
14208:
14186:
14182:
14177:
14173:
14149:
14144:
14140:
14135:
14131:
14128:
14123:
14120:
14115:
14109:
14106:
14065:
14062:
14048:
14043:
14040:
14036:
14032:
14028:
14003:
13998:
13995:
13991:
13987:
13983:
13928:
13927:
13923:
13911:
13908:
13904:
13901:
13897:
13894:
13891:
13887:
13884:
13880:
13877:
13857:
13852:
13830:
13827:
13823:
13818:
13813:
13809:
13803:
13798:
13794:
13790:
13785:
13780:
13776:
13773:
13762:
13746:
13743:
13739:
13736:
13732:
13729:
13726:
13722:
13719:
13715:
13712:
13691:
13686:
13683:
13679:
13675:
13671:
13662:is exact over
13651:
13648:
13644:
13639:
13636:
13632:
13628:
13623:
13620:
13616:
13612:
13607:
13604:
13600:
13596:
13592:
13589:
13585:
13580:
13577:
13573:
13569:
13564:
13561:
13557:
13554:
13543:
13524:
13520:
13515:
13512:
13508:
13504:
13500:
13497:
13492:
13489:
13484:
13480:
13475:
13472:
13468:
13464:
13460:
13450:
13439:
13435:
13430:
13427:
13423:
13419:
13415:
13402:disjoint from
13386:
13381:
13378:
13374:
13370:
13364:
13359:
13354:
13319:
13316:
13311:
13307:
13302:
13297:
13294:
13290:
13286:
13282:
13279:
13275:
13270:
13267:
13263:
13259:
13255:
13243:is given by a
13214:
13211:
13206:
13201:
13198:
13175:
13170:
13147:
13141:
13137:
13111:
13081:
13075:
13070:
13064:
13032:
13027:
13007:
13003:
12998:
12995:
12991:
12987:
12983:
12960:
12955:
12934:
12929:
12924:
12921:
12918:
12915:
12884:
12881:
12878:
12875:
12872:
12869:
12866:
12862:
12858:
12855:
12852:
12849:
12846:
12842:
12837:
12834:
12830:
12826:
12822:
12802:
12799:
12796:
12792:
12789:
12786:
12783:
12779:
12774:
12770:
12765:
12761:
12740:
12735:
12732:
12728:
12724:
12720:
12707:an element in
12687:is called the
12675:
12670:
12667:
12663:
12659:
12655:
12635:
12631:
12626:
12623:
12619:
12615:
12611:
12583:
12579:
12574:
12571:
12567:
12563:
12559:
12530:
12525:
12522:
12518:
12514:
12510:
12507:
12504:
12484:
12479:
12476:
12472:
12468:
12465:
12433:
12430:
12403:
12394:
12383:
12372:
12360:
12354:consisting of
12340:
12336:
12332:
12309:
12305:
12301:
12296:
12293:
12262:
12253:
12242:
12233:
12224:
12213:
12204:
12183:
12174:
12157:
12150:filtered limit
12135:
12134:
12104:
12101:
12098:
12095:
12092:
12087:
12084:
12063:
12051:
12048:
12044:
12041:
12038:
12034:
12031:
12008:
12004:
12001:
11998:
11994:
11991:
11986:
11983:
11979:
11975:
11970:
11967:
11962:
11959:
11955:
11952:
11949:
11945:
11942:
11923:
11912:
11905:
11901:
11895:
11890:
11885:
11882:
11877:
11872:
11866:
11862:
11839:
11828:
11825:
11820:
11816:
11812:
11809:
11806:
11801:
11797:
11793:
11788:
11784:
11780:
11777:
11774:
11769:
11766:
11761:
11758:
11755:
11752:
11747:
11743:
11739:
11734:
11730:
11726:
11723:
11699:
11672:
11656:
11652:
11648:
11643:
11640:
11625:
11607:
11598:
11591:
11585:
11582:
11569:
11551:
11525:
11505:
11504:
11493:
11490:
11487:
11482:
11478:
11474:
11471:
11466:
11462:
11458:
11455:
11448:
11444:
11439:
11433:
11428:
11425:
11422:
11418:
11414:
11411:
11408:
11405:
11402:
11397:
11393:
11369:
11364:
11360:
11342:
11324:
11320:
11316:
11311:
11307:
11301:
11296:
11293:
11290:
11286:
11282:
11279:
11259:
11256:-module, then
11231:
11221:
11206:
11205:
11194:
11191:
11186:
11183:
11179:
11175:
11172:
11169:
11165:
11162:
11159:
11156:
11153:
11148:
11144:
11140:
11137:
11134:
11129:
11124:
11120:
11116:
11113:
11108:
11104:
11080:
11070:
11048:
11043:
11038:
11034:
11030:
11027:
11022:
11017:
11013:
11009:
11006:
10986:
10983:
10980:
10977:
10974:
10969:
10965:
10961:
10958:
10953:
10949:
10945:
10942:
10937:
10933:
10909:
10871:
10839:
10836:
10817:
10816:
10805:
10802:
10797:
10793:
10788:
10783:
10779:
10773:
10769:
10763:
10758:
10754:
10730:
10726:
10723:
10720:
10716:
10713:
10678:
10672:
10668:
10664:
10661:
10658:
10653:
10649:
10644:
10640:
10564:
10561:
10540:
10539:
10526:
10523:
10518:
10515:
10511:
10508:
10505:
10502:
10499:
10496:
10493:
10427:
10424:
10402:
10399:
10396:
10393:
10390:
10385:
10382:
10360:
10357:
10354:
10351:
10348:
10345:
10342:
10337:
10334:
10304:
10301:
10298:
10295:
10292:
10151:
10150:
10139:
10135:
10129:
10125:
10121:
10116:
10112:
10107:
10103:
10100:
10097:
10093:
10090:
10087:
10084:
10081:
10078:
10075:
10072:
10069:
10066:
10063:
10060:
10036:
10030:
10026:
10022:
10017:
10013:
10008:
10004:
9974:is denoted by
9961:
9958:
9955:
9952:
9949:
9946:
9882:(that is, the
9880:
9879:
9868:
9865:
9862:
9859:
9856:
9853:
9849:
9846:
9843:
9840:
9837:
9834:
9831:
9800:
9797:
9794:
9719:
9714:
9710:
9706:
9702:
9680:
9676:
9672:
9669:
9666:
9661:
9657:
9635:
9629:
9625:
9621:
9618:
9615:
9610:
9606:
9601:
9597:
9574:
9573:
9561:
9557:
9554:
9549:
9545:
9541:
9538:
9535:
9532:
9529:
9524:
9520:
9516:
9513:
9508:
9504:
9500:
9497:
9494:
9489:
9486:
9483:
9479:
9473:
9470:
9467:
9463:
9459:
9454:
9450:
9444:
9440:
9435:
9431:
9428:
9425:
9422:
9419:
9393:Main article:
9390:
9387:
9362:
9348:
9343:
9339:
9335:
9332:
9327:
9321:
9293:
9289:
9281:
9267:
9262:
9256:
9233:
9228:
9222:
9216:
9211:
9207:
9202:
9197:
9193:
9189:
9186:
9183:
9178:
9174:
9170:
9167:
9154:. Assume that
9133:
9127:
9121:
9116:
9111:
9105:
9098:
9095:
9092:
9089:
9086:
9083:
9080:
9075:
9069:
9061:
9055:
9048:
9043:
9037:
9031:
9028:
9025:
9020:
9014:
9008:
9005:
8992:are ideals of
8979:
8973:
8948:
8942:
8917:
8911:
8886:
8882:
8878:
8875:
8872:
8869:
8864:
8860:
8839:
8836:
8833:
8830:
8827:
8824:
8819:
8815:
8792:
8789:
8784:
8777:
8772:
8767:
8764:
8761:
8758:
8753:
8746:
8741:
8736:
8733:
8730:
8722:
8716:
8708:
8703:
8700:
8697:
8693:
8687:
8683:
8679:
8673:
8667:
8660:
8656:
8650:
8645:
8642:
8639:
8635:
8631:
8623:
8617:
8608:
8603:
8600:
8597:
8593:
8586:
8582:
8548:
8545:
8542:
8539:
8534:
8528:
8522:
8517:
8511:
8486:
8480:
8474:
8471:
8468:
8463:
8457:
8425:
8421:
8415:
8412:
8409:
8405:
8387:
8373:direct product
8370:is called the
8348:
8341:
8328:
8321:
8314:
8313:
8298:
8293:
8289:
8285:
8280:
8276:
8272:
8267:
8263:
8259:
8254:
8250:
8246:
8243:
8240:
8235:
8231:
8227:
8222:
8218:
8214:
8211:
8208:
8203:
8199:
8195:
8190:
8186:
8182:
8179:
8177:
8175:
8172:
8167:
8163:
8159:
8154:
8150:
8146:
8141:
8137:
8133:
8128:
8124:
8120:
8117:
8114:
8109:
8105:
8101:
8096:
8092:
8088:
8085:
8082:
8077:
8073:
8069:
8064:
8060:
8056:
8053:
8051:
8049:
8010:Main article:
8007:
8006:Direct product
8004:
8002:
7999:
7829:noncommutative
7825:
7824:
7809:
7806:
7803:
7800:
7797:
7794:
7791:
7788:
7785:
7782:
7779:
7776:
7774:
7772:
7769:
7766:
7763:
7760:
7757:
7755:
7752:
7749:
7746:
7743:
7740:
7737:
7734:
7731:
7728:
7725:
7722:
7719:
7716:
7714:
7712:
7709:
7706:
7703:
7700:
7697:
7694:
7691:
7688:
7685:
7682:
7679:
7676:
7673:
7671:
7661:
7550:Main article:
7547:
7544:
7517:
7514:
7511:
7508:
7505:
7501:
7497:
7494:
7489:
7486:
7449:
7448:
7437:
7434:
7431:
7426:
7423:
7418:
7415:
7395:
7392:
7389:
7385:
7381:
7378:
7373:
7370:
7265:
7264:
7249:
7246:
7243:
7240:
7237:
7234:
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7193:
7191:
7188:
7185:
7182:
7179:
7176:
7173:
7170:
7167:
7164:
7161:
7158:
7155:
7152:
7149:
7146:
7143:
7140:
7137:
7134:
7131:
7128:
7126:
7124:
7104:is the set of
7046:quotient group
7040:The notion of
7034:Main article:
7031:
7028:
6975:is called the
6955:
6954:
6935:
6920:
6916:
6912:
6894:
6873:
6870:
6866:
6848:
6823:
6792:
6770:
6767:
6763:
6759:
6756:
6753:
6750:
6747:
6744:
6741:
6738:
6719:
6704:
6700:
6696:
6691:
6665:
6607:
6606:
6589:
6585:
6581:
6578:
6573:
6569:
6565:
6562:
6559:
6557:
6554:
6551:
6548:
6545:
6542:
6539:
6536:
6533:
6530:
6527:
6524:
6521:
6518:
6515:
6512:
6509:
6506:
6504:
6501:
6498:
6495:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6468:
6465:
6462:
6459:
6456:
6453:
6451:
6411:is a function
6379:Main article:
6376:
6373:
6325:Equivalently,
6314:
6311:
6308:
6305:
6285:
6282:
6279:
6259:
6256:
6253:
6250:
6230:
6227:
6224:
6221:
6218:
6148:is written as
6035:
6034:
6023:
6020:
6017:
6012:
6008:
5995:
5992:
5987:
5983:
5960:
5956:
5950:
5946:
5942:
5939:
5936:
5931:
5927:
5921:
5917:
5842:be a ring. A
5832:Main article:
5829:
5826:
5685:characteristic
5682:is called the
5647:is called the
5584:
5580:
5556:
5552:
5522:
5518:
5493:
5468:
5465:
5462:
5458:
5425:
5411:
5410:
5394:
5363:; and for all
5355:
5303:Main article:
5300:
5297:
5231:is an element
5199:
5158:is an element
5132:is an element
5121:
5118:
5041:
5032:
5023:
5013:
4995:
4991:
4985:
4980:
4977:
4974:
4970:
4966:
4961:
4957:
4928:
4923:
4919:
4915:
4912:
4909:
4904:
4900:
4896:
4880:
4877:
4875:
4874:Basic concepts
4872:
4871:
4870:
4846:
4843:
4840:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4804:
4799:
4796:
4792:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4745:
4723:
4719:
4691:is not even a
4678:
4675:
4672:
4668:
4664:
4639:
4620:
4617:
4616:
4615:
4604:
4543:
4536:
4524:
4466:
4416:
4413:
4412:
4411:
4384:
4377:
4357:
4352:
4322:
4316:
4312:
4292:
4274:
4270:
4244:
4226:
4225:
4224:
4201:
4182:
4160:
4140:
4137:
4132:
4128:
4124:
4121:
4118:
4113:
4109:
4105:
4102:
4099:
4088:
4054:
4047:
4042:
4039:
4031:
4030:Basic examples
4028:
4027:
4026:
4025:
4024:
4021:
4008:direct product
3972:
3969:
3949:Adolf Fraenkel
3944:
3941:
3915:
3914:
3899:
3895:
3893:
3888:
3885:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3861:
3857:
3853:
3850:
3847:
3844:
3842:
3838:
3834:
3830:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3806:
3802:
3798:
3795:
3792:
3790:
3786:
3782:
3778:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3754:
3750:
3746:
3743:
3740:
3738:
3734:
3730:
3726:
3725:
3722:
3719:
3716:
3711:
3707:
3703:
3700:
3697:
3695:
3691:
3687:
3683:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3659:
3655:
3651:
3647:
3646:
3611:
3608:
3587:
3584:
3562:
3559:
3521:
3517:
3509:
3506:
3504:
3501:
3500:
3497:
3494:
3492:
3489:
3488:
3484:
3480:
3477:
3474:
3453:
3445:
3442:
3440:
3437:
3436:
3433:
3430:
3428:
3425:
3424:
3420:
3416:
3413:
3410:
3390:
3386:
3378:
3375:
3373:
3370:
3369:
3366:
3363:
3361:
3358:
3357:
3353:
3349:
3346:
3325:
3317:
3314:
3312:
3309:
3308:
3305:
3302:
3300:
3297:
3296:
3292:
3288:
3285:
3264:
3256:
3253:
3251:
3248:
3247:
3244:
3241:
3239:
3236:
3235:
3231:
3210:
3207:
3204:
3201:
3196:
3192:
3176:
3175:
3164:
3160:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3125:
3120:
3114:
3111:
3109:
3106:
3105:
3102:
3099:
3097:
3094:
3093:
3091:
3086:
3082:
3078:
3075:
3072:
3069:
3066:
3061:
3057:
3030:
3027:
3015:
3010:
3007:
3002:
2997:
2993:
2990:
2984:
2979:
2976:
2971:
2951:
2946:
2942:
2939:
2933:
2928:
2925:
2920:
2897:
2893:
2889:
2884:
2859:
2856:
2830:
2825:
2822:
2791:
2787:
2783:
2779:
2774:
2737:
2733:
2707:
2703:
2699:
2694:
2680:
2679:
2668:
2663:
2660:
2655:
2650:
2647:
2642:
2637:
2634:
2610:
2607:
2602:
2597:
2594:
2589:
2584:
2581:
2566:is divided by
2548:
2544:
2540:
2535:
2510:
2507:
2502:
2497:
2494:
2481:
2470:
2465:
2462:
2457:
2452:
2449:
2444:
2439:
2436:
2412:
2409:
2404:
2399:
2396:
2391:
2386:
2383:
2333:is divided by
2309:
2305:
2301:
2296:
2271:
2268:
2263:
2258:
2255:
2228:
2222:
2219:
2214:
2209:
2206:
2201:
2196:
2193:
2188:
2183:
2180:
2174:
2170:
2166:
2162:
2158:
2153:
2143:Equip the set
2134:
2131:
2130:
2129:
2110:holds for any
2104:
2093:
2070:
2034:
2031:
2028:
2020:
2017:
2013:
2012:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1886:
1882:
1864:multiplication
1847:
1844:
1760:Β§ History
1725:
1722:
1714:is written as
1698:
1697:
1696:
1695:
1643:
1585:
1584:
1583:
1535:
1473:
1472:
1471:
1423:
1386:
1344:
1258:
1255:
1229:that occur in
1087:multiplication
1073:
1072:
1070:
1069:
1062:
1055:
1047:
1044:
1043:
1039:
1038:
1033:
1028:
1023:
1018:
1013:
1008:
1002:
1001:
1000:
994:
988:
987:
984:
983:
980:
979:
976:Linear algebra
970:
969:
964:
959:
953:
952:
946:
945:
942:
941:
938:
937:
934:Lattice theory
930:
923:
922:
917:
912:
907:
902:
897:
891:
890:
884:
883:
880:
879:
870:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
818:
817:
811:
810:
807:
806:
797:
796:
791:
786:
780:
779:
778:
773:
768:
759:
753:
747:
746:
743:
742:
703:multiplication
657:
656:
654:
653:
646:
639:
631:
628:
627:
619:
618:
590:
589:
583:
577:
571:
559:
554:
553:
550:
549:
546:
545:
534:
529:
525:
521:
517:
498:
485:
480:
461:
448:
443:
431:-adic integers
424:
418:
409:
395:
394:
393:
392:
386:
380:
379:
378:
366:
360:
354:
348:
342:
324:
319:
318:
315:
314:
311:
310:
309:
308:
296:
295:
294:
288:
276:
275:
274:
256:
255:
254:
253:
241:
237:
233:
228:
224:
221:
207:
195:
174:
168:
162:
156:
142:
141:
135:
129:
115:
114:
108:
102:
96:
95:
94:
88:
76:
70:
58:
56:Basic concepts
55:
54:
51:
50:
26:
9:
6:
4:
3:
2:
21656:
21645:
21642:
21640:
21637:
21636:
21634:
21622:
21618:
21614:
21609:
21604:
21600:
21596:
21595:
21590:
21586:
21582:
21578:
21574:
21570:
21566:
21562:
21558:
21557:
21552:
21548:
21545:
21541:
21537:
21534:
21533:0-8218-0993-8
21530:
21527:
21523:
21519:
21515:
21511:
21507:
21503:
21501:
21498:
21495:
21494:3-540-43491-7
21491:
21487:
21483:
21482:
21470:
21466:
21462:
21458:
21454:
21450:
21449:Math. Annalen
21446:
21442:
21441:Noether, Emmy
21438:
21434:
21430:
21426:
21422:
21418:
21414:
21410:
21406:
21402:
21398:
21394:
21390:
21389:
21377:
21375:0-387-90089-6
21371:
21367:
21363:
21359:
21355:
21352:
21347:
21345:9780821891322
21341:
21337:
21336:
21330:
21327:
21325:9783540373704
21321:
21317:
21316:
21310:
21306:
21302:
21298:
21295:
21291:
21286:
21281:
21277:
21273:
21269:
21265:
21264:Poonen, Bjorn
21261:
21257:
21255:0-387-90693-2
21251:
21247:
21246:
21240:
21237:
21233:
21229:
21223:
21219:
21215:
21211:
21207:
21202:
21198:
21196:9780486411477
21192:
21188:
21187:
21182:
21179:Korn, G. A.;
21177:
21173:
21169:
21165:
21162:
21158:
21154:
21150:
21146:
21142:
21138:
21134:
21133:
21128:
21124:
21120:
21114:
21110:
21106:
21102:
21101:Isaacs, I. M.
21098:
21094:
21089:
21084:
21079:
21075:
21071:
21067:
21062:
21059:
21057:9780521432177
21053:
21048:
21047:
21041:
21037:
21033:
21028:
21024:
21020:
21015:
21012:
21006:
21002:
20997:
20994:
20988:
20984:
20979:
20978:
20966:
20961:
20957:
20952:
20948:
20946:9780486663418
20942:
20938:
20933:
20930:
20926:
20922:
20916:
20912:
20911:
20906:
20902:
20899:
20897:0-387-98541-7
20893:
20889:
20888:Galois Theory
20884:
20880:
20875:
20871:
20865:
20861:
20857:
20852:
20848:
20844:
20840:
20837:
20835:0-387-22025-9
20831:
20827:
20822:
20819:
20815:
20811:
20807:
20803:
20797:
20793:
20789:
20785:
20781:
20777:
20775:0-387-00500-5
20771:
20767:
20763:
20759:
20755:
20753:0-387-95183-0
20749:
20745:
20741:
20737:
20733:
20731:0-387-98428-3
20727:
20723:
20719:
20715:
20712:
20708:
20704:
20702:0-226-42454-5
20698:
20694:
20690:
20689:
20684:
20680:
20676:
20670:
20666:
20665:Basic algebra
20662:
20658:
20654:
20650:
20646:
20642:
20638:
20634:
20630:
20626:
20622:
20620:9780030105593
20616:
20612:
20607:
20603:
20601:0-88385-015-X
20597:
20593:
20589:
20585:
20581:
20575:
20571:
20566:
20562:
20560:9780618514717
20556:
20552:
20547:
20543:
20539:
20535:
20531:
20527:
20524:
20518:
20514:
20510:
20506:
20502:
20498:
20494:
20490:
20486:
20482:
20478:
20474:
20470:
20466:
20462:
20458:
20454:
20453:
20441:
20436:
20433:
20432:
20426:
20423:
20417:
20413:
20408:
20404:
20398:
20394:
20389:
20388:
20374:
20368:
20363:
20356:
20351:
20345:
20340:
20333:
20328:
20321:
20317:
20316:Weibel (2013)
20312:
20305:
20300:
20294:
20290:
20285:
20278:
20273:
20266:
20261:
20254:
20249:
20242:
20237:
20230:
20225:
20218:
20213:
20206:
20201:
20199:
20191:
20186:
20179:
20174:
20167:
20162:
20155:
20150:
20144:, p. 155
20143:
20138:
20131:
20130:Rotman (1998)
20126:
20120:, p. 176
20119:
20118:Wilder (1965)
20114:
20108:
20107:Poonen (2019)
20103:
20097:
20092:
20084:
20080:
20074:
20068:
20063:
20057:, p. 188
20056:
20055:Warner (1965)
20051:
20044:
20039:
20033:, p. 235
20032:
20027:
20020:
20015:
20008:
20003:
19997:, p. 346
19996:
19991:
19985:
19980:
19974:
19969:
19960:
19956:
19951:
19944:
19939:
19937:
19927:
19923:
19918:
19911:
19906:
19899:
19894:
19888:
19884:
19879:
19873:
19868:
19861:
19856:
19848:
19842:
19836:, p. 158
19835:
19830:
19823:
19818:
19811:
19806:
19800:, Theorem 3.1
19799:
19794:
19788:, p. 161
19787:
19786:Isaacs (1994)
19782:
19774:
19770:
19764:
19758:, p. 160
19757:
19756:Isaacs (1994)
19752:
19745:
19740:
19738:
19730:
19725:
19718:
19713:
19709:
19696:
19690:
19681:
19668:
19661:
19657:
19656:
19649:
19632:
19628:
19617:
19613:
19610:
19608:
19605:
19603:
19600:
19598:
19595:
19593:
19590:
19588:
19585:
19583:
19580:
19578:
19574:
19571:
19569:
19566:
19564:
19561:
19559:
19556:
19554:
19551:
19549:
19546:
19544:
19543:Dedekind ring
19541:
19539:
19536:
19535:
19533:
19527:
19524:
19522:
19519:
19517:
19514:
19512:
19509:
19507:
19504:
19502:
19499:
19497:
19494:
19492:
19489:
19487:
19484:
19483:
19476:
19474:
19470:
19469:monoid object
19463:
19458:
19454:
19437:
19431:
19428:
19425:
19422:
19419:
19408:
19404:
19403:ring spectrum
19400:
19393:Ring spectrum
19390:
19380:
19370:
19365:
19335:
19332:
19320:
19306:
19302:
19292:
19279:
19222:
19218:
19178:
19174:
19170:
19164:
19161:
19158:
19152:
19149:
19146:
19141:
19137:
19112:
19104:
19092:
19072:
19064:
19051:
19031:
19023:
19011:
18991:
18983:
18970:
18967:
18964:
18944:
18936:
18923:
18920:
18917:
18906:is an object
18901:
18893:
18889:
18867:
18863:
18845:
18842:
18839:
18836:
18833:
18830:
18827:
18816:
18815:
18814:
18811:
18805:
18799:
18795:
18787:
18779:
18773:
18769:
18759:
18757:
18753:
18743:
18741:
18731:
18723:
18721:
18717:
18712:
18707:
18701:
18683:
18678:
18641:
18637:
18633:
18629:
18625:
18621:
18606:
18601:
18586:
18585:endomorphisms
18580:
18576:
18569:
18549:
18541:
18537:
18531:
18521:
18517:
18513:
18509:
18481:
18472:
18466:
18460:
18457:
18451:
18442:
18439:
18436:
18431:
18420:
18414:
18411:
18405:
18399:
18396:
18390:
18381:
18378:
18375:
18370:
18358:
18357:
18356:
18353:
18349:
18343:
18339:
18332:
18320:
18304:
18298:
18292:
18284:
18277:
18275:
18267:
18256:
18226:
18222:
18218:
18214:
18208:
18198:
18196:
18192:
18188:
18184:
18174:
18172:
18168:
18164:
18160:
18156:
18152:
18148:
18138:
18136:
18132:
18127:
18123:
18119:
18109:
18107:
18102:
18098:
18094:
18093:Burnside ring
18090:
18080:
18078:
18074:
18070:
18069:fiber bundles
18066:
18061:
18058:
18054:
18045:
18037:
18033:
18029:
18025:
18021:
18017:
18013:
17989:
17986:
17978:
17974:
17966:
17962:
17943:
17932:
17929:
17921:
17917:
17906:
17903:
17900:
17896:
17892:
17881:
17878:
17870:
17866:
17858:
17857:
17856:
17855:
17848:
17838:
17836:
17832:
17819:
17815:
17811:
17810:
17806:
17802:
17786:
17775:
17772:
17761:
17755:
17747:
17743:
17709:For example,
17708:
17707:
17690:
17684:
17676:
17673:
17670:
17666:
17659:
17651:
17647:
17641:
17636:
17632:
17628:
17622:
17619:
17616:
17608:
17604:
17596:
17595:
17593:
17584:
17580:
17576:
17567:
17563:
17560:
17556:
17544:
17541:inherits the
17539:
17535:
17530:
17525:
17521:
17517:
17499:
17493:
17490:
17487:
17484:
17481:
17473:
17465:
17457:
17453:
17411:over a field
17410:
17406:
17402:
17401:
17400:
17398:
17397:abelian group
17387:
17383:
17373:
17369:
17344:
17334:
17330:
17309:
17303:
17300:
17297:
17291:
17285:
17279:
17274:
17271:
17268:
17264:
17260:
17254:
17245:
17242:
17239:
17228:
17224:
17193:
17183:
17167:
17153:
17143:
17133:
17126:
17112:
17108:
17096:over a field
17076:
17066:
17059:
17056:The field of
17055:
17054:
17053:
17050:
17046:
17042:
17025:
17019:
17015:
17011:
17007:
17003:
16999:
16995:
16989:
16985:
16975:
16969:
16959:
16951:
16941:
16931:
16929:
16925:
16912:
16908:
16902:
16898:
16894:
16888:
16884:
16880:
16874:
16871:
16867:
16861:
16857:
16846:
16842:
16812:
16807:
16803:
16799:
16789:
16785:
16777:
16773:
16755:
16750:
16746:
16742:
16725:
16723:
16722:invariant map
16701:
16692:
16686:
16680:
16677:
16651:
16633:
16625:
16598:
16595:
16587:
16583:
16573:
16568:For example,
16566:
16564:
16558:
16548:
16531:
16527:
16522:
16518:
16514:
16510:
16506:
16500:
16491:
16468:
16463:
16460:
16456:
16452:
16447:
16443:
16437:
16433:
16427:
16423:
16402:
16397:
16393:
16387:
16383:
16379:
16376:
16371:
16367:
16361:
16357:
16353:
16337:
16324:
16322:
16317:
16313:
16309:
16284:
16266:
16256:
16254:
16245:
16241:
16237:
16221:
16216:
16212:
16208:
16200:
16193:over a field
16184:
16178:
16174:
16167:
16163:
16141:
16137:
16130:
16123:
16119:
16108:
16103:
16100:
16097:
16093:
16085:
16078:
16075:
16074:semiprimitive
16071:
16064:
16058:
16057:
16056:
16049:
16041:
16039:
16035:
16031:
16023:
16020:
16017:
16009:
16001:
15996:
15983:
15980:
15974:
15969:
15954:
15951:
15947:
15946:division ring
15943:
15942:
15936:
15934:
15933:
15928:
15927:
15920:
15910:
15908:
15904:
15903:L. E. Dickson
15900:
15895:
15893:
15888:
15885:
15883:
15879:
15875:
15871:
15867:
15866:division ring
15860:Division ring
15855:
15854:
15849:
15848:
15843:
15842:
15837:
15836:
15831:
15830:
15825:
15824:
15819:
15818:
15813:
15812:
15807:
15806:
15801:
15800:
15795:
15794:
15790:
15789:
15788:
15786:
15781:
15779:
15770:
15766:
15760:
15755:
15748:
15732:
15731:Jordan matrix
15705:
15700:
15696:
15692:
15689:
15686:
15680:
15672:
15668:
15647:
15643:
15636:
15632:
15626:
15622:
15618:
15613:
15606:
15600:
15592:
15583:
15562:
15556:
15550:
15547:
15544:
15541:
15538:
15529:
15516:
15509:
15505:
15499:
15495:
15491:
15484:
15480:
15474:
15470:
15466:
15463:
15450:
15440:
15436:
15432:
15419:
15414:
15412:
15408:
15404:
15400:
15396:
15392:
15388:
15384:
15380:
15379:zero-divisors
15376:
15360:
15356:
15351:
15338:
15335:
15332:
15329:
15326:
15323:
15320:
15314:
15311:
15308:
15299:
15296:
15293:
15266:
15261:
15257:
15253:
15236:
15230:
15224:
15222:
15218:
15214:
15198:
15187:
15178:
15173:with symbols
15168:
15144:
15138:
15135:
15132:
15129:
15126:
15123:
15120:
15117:
15114:
15111:
15108:
15102:
15099:
15073:
15025:
15022:
15018:
14989:
14975:
14973:
14968:
14964:
14959:
14957:
14956:
14949:
14942:
14935:
14929:
14914:
14909:
14889:
14883:
14856:
14825:
14820:
14803:
14733:
14723:
14714:
14708:
14698:
14690:
14685:
14675:
14671:
14667:
14642:
14639:
14634:
14624:
14588:
14580:
14576:
14550:
14542:
14538:
14534:
14530:
14526:
14521:
14511:
14429:
14418:
14366:
14351:
14330:
14325:
14308:
14295:
14262:
14256:
14224:
14215:
14206:
14184:
14180:
14175:
14171:
14147:
14142:
14138:
14133:
14129:
14126:
14121:
14118:
14113:
14104:
14085:
14084:
14061:
14046:
14041:
14038:
14034:
14030:
14026:
14001:
13996:
13993:
13989:
13985:
13981:
13974:then maps to
13961:
13937:
13933:
13924:
13909:
13902:
13899:
13892:
13885:
13882:
13875:
13855:
13828:
13821:
13811:
13796:
13788:
13778:
13771:
13763:
13744:
13737:
13734:
13727:
13720:
13717:
13710:
13689:
13684:
13681:
13677:
13673:
13669:
13649:
13642:
13637:
13634:
13630:
13626:
13621:
13618:
13610:
13605:
13602:
13598:
13594:
13590:
13583:
13578:
13575:
13571:
13567:
13562:
13559:
13552:
13544:
13522:
13518:
13513:
13510:
13506:
13502:
13498:
13495:
13490:
13487:
13482:
13478:
13473:
13470:
13466:
13462:
13458:
13451:
13437:
13433:
13428:
13425:
13421:
13417:
13413:
13384:
13379:
13376:
13372:
13368:
13343:
13342:
13341:
13330:
13317:
13314:
13309:
13305:
13300:
13295:
13292:
13288:
13284:
13280:
13277:
13273:
13268:
13265:
13261:
13257:
13253:
13246:
13225:
13212:
13196:
13168:
13139:
13135:
13079:
13068:
13053:
13052:maximal ideal
13049:
13025:
13005:
13001:
12996:
12993:
12989:
12985:
12981:
12953:
12932:
12922:
12919:
12916:
12913:
12896:
12882:
12876:
12873:
12870:
12867:
12860:
12853:
12847:
12844:
12840:
12835:
12832:
12828:
12824:
12820:
12800:
12797:
12794:
12790:
12787:
12784:
12781:
12777:
12772:
12768:
12763:
12759:
12738:
12733:
12730:
12726:
12722:
12718:
12690:
12673:
12668:
12665:
12661:
12657:
12653:
12633:
12629:
12624:
12621:
12617:
12613:
12609:
12581:
12577:
12572:
12569:
12565:
12561:
12557:
12528:
12523:
12520:
12516:
12512:
12508:
12502:
12477:
12474:
12470:
12463:
12448:and a subset
12443:
12439:
12429:
12427:
12426:
12420:
12417:
12413:
12406:
12402:
12397:
12393:
12386:
12382:
12375:
12371:
12363:
12359:
12338:
12334:
12330:
12307:
12303:
12299:
12294:
12291:
12280:
12276:
12272:
12265:
12261:
12256:
12252:
12245:
12241:
12236:
12232:
12227:
12223:
12216:
12212:
12207:
12203:
12197:
12193:
12186:
12182:
12177:
12173:
12167:
12160:
12156:
12151:
12147:
12142:
12140:
12132:
12131:generic point
12128:
12119:
12099:
12093:
12090:
12085:
12082:
12069:over a field
12068:
12064:
12049:
12039:
12029:
12022:
11999:
11989:
11984:
11981:
11977:
11973:
11968:
11965:
11960:
11950:
11940:
11929:over a field
11928:
11925:The field of
11924:
11910:
11903:
11899:
11888:
11883:
11880:
11875:
11870:
11848:
11847:finite fields
11844:
11840:
11826:
11818:
11814:
11810:
11807:
11804:
11799:
11795:
11791:
11786:
11782:
11775:
11772:
11767:
11764:
11759:
11753:
11750:
11745:
11741:
11737:
11732:
11728:
11721:
11713:
11712:
11711:
11708:
11694:
11690:
11684:
11680:
11654:
11650:
11646:
11641:
11638:
11621:
11610:
11606:
11581:
11577:
11572:
11559:
11554:
11543:
11539:
11533:
11528:
11516:
11514:
11510:
11491:
11480:
11476:
11469:
11464:
11460:
11453:
11446:
11442:
11431:
11426:
11423:
11420:
11416:
11412:
11406:
11400:
11395:
11391:
11383:
11382:
11381:
11367:
11362:
11358:
11322:
11318:
11314:
11309:
11305:
11299:
11294:
11291:
11288:
11284:
11280:
11277:
11267:
11262:
11248:says that if
11247:
11246:Schur's lemma
11243:
11239:
11234:
11229:
11224:
11216:
11212:
11192:
11184:
11181:
11177:
11167:
11163:
11157:
11151:
11146:
11132:
11127:
11122:
11118:
11111:
11106:
11102:
11094:
11093:
11092:
11088:
11083:
11078:
11046:
11041:
11036:
11032:
11025:
11020:
11015:
11011:
11007:
11004:
10984:
10978:
10972:
10967:
10959:
10951:
10947:
10940:
10935:
10931:
10921:
10917:
10912:
10901:
10879:
10874:
10867:
10849:
10845:
10835:
10833:
10829:
10823:
10803:
10800:
10795:
10791:
10786:
10781:
10777:
10771:
10767:
10756:
10752:
10744:
10743:
10742:
10721:
10711:
10704:
10699:
10697:
10696:GrΓΆbner basis
10676:
10670:
10666:
10662:
10659:
10656:
10651:
10647:
10642:
10638:
10630:
10621:
10618:
10600:
10590:
10589:is infinite.
10580:
10559:
10521:
10513:
10509:
10506:
10497:
10491:
10484:
10483:
10482:
10479:
10475:
10449:
10447:
10440:restricts to
10422:
10400:
10397:
10391:
10380:
10358:
10349:
10343:
10340:
10332:
10302:
10296:
10293:
10290:
10281:
10269:
10265:
10259:
10255:
10251:
10247:
10243:
10239:
10223:
10219:
10215:
10211:
10207:
10201:
10194:
10190:
10186:
10174:Example: let
10172:
10169:
10163:
10159:generated by
10157:
10137:
10133:
10127:
10123:
10119:
10114:
10110:
10105:
10101:
10095:
10091:
10085:
10079:
10070:
10067:
10064:
10058:
10051:
10050:
10049:
10034:
10028:
10024:
10020:
10015:
10011:
10006:
10002:
9993:
9984:generated by
9978:
9956:
9950:
9944:
9932:
9928:
9918:
9914:
9910:
9904:
9900:
9894:
9890:
9885:
9863:
9857:
9851:
9847:
9844:
9835:
9829:
9822:
9821:
9820:
9798:
9795:
9792:
9783:
9780:
9770:
9760:
9751:
9746:
9741:
9732:
9731:as subrings.
9717:
9712:
9708:
9704:
9700:
9678:
9674:
9670:
9667:
9664:
9659:
9655:
9633:
9627:
9623:
9619:
9616:
9613:
9608:
9604:
9599:
9595:
9583:
9559:
9555:
9552:
9547:
9543:
9539:
9536:
9533:
9530:
9527:
9522:
9518:
9514:
9511:
9506:
9502:
9498:
9495:
9492:
9487:
9484:
9481:
9477:
9471:
9468:
9465:
9461:
9457:
9452:
9448:
9442:
9438:
9433:
9429:
9423:
9417:
9410:
9409:
9408:
9396:
9386:
9384:
9380:
9376:
9371:
9346:
9341:
9337:
9333:
9330:
9325:
9307:
9303:
9296:
9278:one has that
9265:
9260:
9231:
9226:
9214:
9209:
9205:
9200:
9195:
9191:
9187:
9184:
9181:
9176:
9172:
9168:
9165:
9153:
9131:
9119:
9114:
9109:
9096:
9093:
9090:
9087:
9084:
9081:
9078:
9073:
9059:
9046:
9041:
9029:
9026:
9023:
9018:
9006:
9003:
8977:
8946:
8915:
8884:
8880:
8876:
8873:
8870:
8862:
8858:
8837:
8834:
8831:
8828:
8825:
8822:
8817:
8813:
8803:
8790:
8782:
8765:
8762:
8759:
8756:
8751:
8734:
8720:
8706:
8701:
8698:
8695:
8691:
8681:
8677:
8671:
8658:
8654:
8648:
8643:
8640:
8637:
8633:
8629:
8621:
8606:
8601:
8598:
8595:
8591:
8584:
8580:
8572:
8567:
8563:
8543:
8537:
8532:
8520:
8515:
8484:
8472:
8469:
8466:
8461:
8439:
8423:
8419:
8413:
8410:
8407:
8403:
8375:
8374:
8364:
8360:
8347:
8340:
8327:
8320:
8291:
8287:
8283:
8278:
8274:
8270:
8265:
8261:
8257:
8252:
8248:
8241:
8233:
8229:
8225:
8220:
8216:
8209:
8201:
8197:
8193:
8188:
8184:
8178:
8165:
8161:
8157:
8152:
8148:
8144:
8139:
8135:
8131:
8126:
8122:
8115:
8107:
8103:
8099:
8094:
8090:
8083:
8075:
8071:
8067:
8062:
8058:
8052:
8040:
8039:
8038:
8035:
8031:
8027:
8013:
8001:Constructions
7998:
7996:
7980:
7974:
7970:
7960:
7956:
7952:
7948:
7934:
7930:
7926:
7920:
7917:
7912:
7908:
7901:
7899:
7895:
7889:
7886:
7882:
7878:
7874:
7870:
7863:
7859:
7855:
7851:
7838:
7837:right modules
7834:
7830:
7804:
7801:
7795:
7792:
7789:
7783:
7780:
7775:
7767:
7764:
7761:
7758:
7750:
7747:
7744:
7741:
7738:
7735:
7732:
7726:
7723:
7720:
7715:
7707:
7704:
7701:
7698:
7695:
7692:
7686:
7683:
7680:
7674:
7662:
7656:
7655:
7654:
7647:
7641:
7631:
7625:
7620:
7603:
7599:
7595:
7591:
7587:
7586:abelian group
7571:
7567:
7563:
7559:
7553:
7543:
7536:
7532:
7515:
7509:
7506:
7503:
7499:
7495:
7492:
7484:
7473:
7469:
7463:
7459:
7455:
7435:
7432:
7429:
7421:
7416:
7413:
7393:
7387:
7383:
7379:
7376:
7368:
7356:
7352:
7346:
7342:
7338:
7333:
7332:
7331:
7328:
7324:
7320:
7314:
7310:
7306:
7302:
7296:
7294:
7289:
7285:
7275:
7271:
7247:
7244:
7241:
7235:
7232:
7226:
7220:
7217:
7214:
7205:
7202:
7199:
7194:
7186:
7183:
7180:
7174:
7171:
7168:
7162:
7156:
7153:
7150:
7144:
7138:
7135:
7132:
7127:
7115:
7114:
7113:
7107:
7102:
7098:
7094:
7093:quotient ring
7088:
7076:
7072:
7063:
7057:
7053:
7047:
7043:
7042:quotient ring
7037:
7036:Quotient ring
7030:Quotient ring
7027:
7013:
6996:
6978:
6969:
6965:
6961:
6952:
6944:
6936:
6899:
6895:
6890:
6871:
6850:For any ring
6849:
6837:
6833:
6828:
6824:
6821:
6811:
6806:
6793:
6786:
6768:
6765:
6761:
6757:
6754:
6748:
6745:
6742:
6736:
6720:
6698:
6694:
6649:(a number in
6640:
6639:
6638:
6635:
6633:
6629:
6621:
6620:
6610:
6587:
6583:
6579:
6571:
6567:
6560:
6549:
6543:
6540:
6534:
6528:
6525:
6519:
6516:
6513:
6507:
6496:
6490:
6487:
6481:
6475:
6472:
6466:
6463:
6460:
6454:
6442:
6441:
6440:
6433:
6427:
6408:
6400:
6396:
6390:
6389:
6382:
6372:
6369:
6365:
6359:
6355:
6349:
6345:
6341:we have that
6339:
6333:
6312:
6309:
6306:
6303:
6283:
6280:
6277:
6257:
6254:
6251:
6248:
6241:we have that
6228:
6225:
6222:
6219:
6216:
6208:
6195:
6193:
6189:
6188:Artinian ring
6185:
6181:
6176:
6174:
6169:
6153:
6146:
6137:
6132:
6126:
6112:
6097:
6080:
6076:
6067:
6063:
6054:
6049:
6045:
6021:
6018:
6015:
6010:
6006:
5993:
5990:
5985:
5981:
5958:
5954:
5948:
5944:
5940:
5937:
5934:
5929:
5925:
5919:
5915:
5907:
5906:
5905:
5882:
5878:
5845:
5835:
5825:
5819:
5807:
5804:, called the
5771:
5768:, called the
5761:
5746:
5742:
5719:
5710:Given a ring
5708:
5706:
5696:
5687:
5686:
5667:
5650:
5637:
5635:
5603:
5601:
5578:
5554:
5516:
5463:
5446:
5442:
5407:
5403:
5395:
5383:
5379:
5361:
5356:
5344:
5340:
5336:
5331:
5324:
5323:
5322:
5320:
5306:
5296:
5294:
5285:
5270:
5266:
5259:
5253:
5248:
5243:
5238:
5230:
5225:
5223:
5218:
5214:
5197:
5190:
5185:
5183:
5179:
5173:
5166:
5157:
5152:
5148:
5127:
5117:
5113:
5109:
5103:
5100:
5096:
5089:
5083:
5080:
5076:
5069:
5059:
5056:
5052:
5044:
5040:
5035:
5031:
5026:
5022:
5012:
4993:
4989:
4983:
4978:
4975:
4972:
4968:
4964:
4959:
4955:
4921:
4917:
4913:
4910:
4907:
4902:
4898:
4864:
4844:
4841:
4838:
4831:
4828:
4825:
4819:
4813:
4807:
4794:
4790:
4786:
4780:
4771:
4768:
4765:
4754:
4746:
4743:
4721:
4698:
4694:
4673:
4670:
4627:
4623:
4622:
4613:
4609:
4605:
4601:
4591:
4585:
4576:
4567:
4562:
4554:
4549:
4544:
4541:
4537:
4532:
4527:
4512:
4503:
4496:
4491:
4486:
4481:
4480:endomorphisms
4477:
4476:abelian group
4472:
4467:
4458:
4447:
4438:
4420:For any ring
4419:
4418:
4409:
4405:
4401:
4393:
4385:
4382:
4378:
4355:
4320:
4314:
4297:
4293:
4290:
4272:
4231:
4227:
4202:
4199:
4169:
4165:
4161:
4154:
4130:
4126:
4122:
4119:
4116:
4111:
4107:
4097:
4089:
4082:
4077:
4072:
4071:
4069:
4059:
4055:
4052:
4048:
4045:
4044:
4037:
4022:
4019:
4018:
4017:
4016:
4015:
4012:
4009:
4005:
3996:
3994:
3976:
3968:
3966:
3962:
3958:
3954:
3950:
3940:
3937:
3931:
3921:
3897:
3894:
3886:
3879:
3876:
3873:
3870:
3867:
3864:
3859:
3855:
3851:
3848:
3845:
3843:
3836:
3832:
3824:
3821:
3818:
3815:
3812:
3809:
3804:
3800:
3796:
3793:
3791:
3784:
3780:
3772:
3769:
3766:
3763:
3760:
3757:
3752:
3748:
3744:
3741:
3739:
3732:
3728:
3720:
3717:
3714:
3709:
3705:
3701:
3698:
3696:
3689:
3685:
3677:
3674:
3671:
3668:
3665:
3662:
3660:
3653:
3649:
3637:
3636:
3635:
3631:
3627:
3622:
3617:
3616:David Hilbert
3607:
3605:
3601:
3597:
3593:
3581:
3577:
3573:
3568:
3558:
3556:
3555:
3533:
3519:
3515:
3507:
3502:
3495:
3490:
3482:
3478:
3475:
3472:
3451:
3443:
3438:
3431:
3426:
3418:
3414:
3411:
3408:
3388:
3384:
3376:
3371:
3364:
3359:
3351:
3347:
3344:
3323:
3315:
3310:
3303:
3298:
3290:
3286:
3283:
3262:
3254:
3249:
3242:
3237:
3229:
3205:
3199:
3194:
3181:
3162:
3158:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3123:
3118:
3112:
3107:
3100:
3095:
3089:
3080:
3076:
3070:
3064:
3059:
3047:
3046:
3045:
3040:
3036:
3026:
3013:
3005:
3000:
2991:
2988:
2982:
2974:
2969:
2962:For example,
2949:
2940:
2937:
2931:
2923:
2918:
2891:
2887:
2854:
2828:
2820:
2808:
2789:
2781:
2777:
2735:
2701:
2697:
2666:
2658:
2653:
2645:
2640:
2632:
2605:
2600:
2592:
2587:
2579:
2542:
2538:
2505:
2500:
2492:
2482:
2468:
2460:
2455:
2447:
2442:
2434:
2407:
2402:
2394:
2389:
2381:
2369:
2365:
2359:
2355:
2345:
2341:
2331:
2327:
2303:
2299:
2266:
2261:
2253:
2243:
2242:
2241:
2226:
2217:
2212:
2204:
2199:
2191:
2186:
2178:
2172:
2168:
2160:
2156:
2140:
2126:
2122:
2109:
2105:
2094:
2091:
2071:
2067:
2063:
2057:
2052:
2048:
2035:
2032:
2029:
2026:
2025:
2024:
2016:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1905:
1904:
1903:
1902:
1884:
1865:
1861:
1857:
1852:
1843:
1842:is a "ring".
1841:
1836:
1833:
1829:
1825:
1821:
1810:
1808:
1804:
1800:
1795:
1793:
1789:
1785:
1784:
1778:
1769:
1764:
1762:
1761:
1756:
1755:even integers
1750:
1732:
1721:
1718:
1712:
1708:
1688:
1684:
1680:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1644:
1636:
1632:
1628:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1592:
1590:
1586:
1581:
1564:
1560:
1553:
1549:
1536:
1524:
1520:
1516:
1509:
1505:
1501:
1497:
1493:
1489:
1484:
1483:
1481:
1474:
1465:
1460:
1452:
1448:
1438:
1433:there exists
1424:
1421:
1404:
1400:
1387:
1384:
1371:
1367:
1361:
1357:
1353:
1349:
1345:
1342:
1329:
1325:
1321:
1314:
1310:
1306:
1302:
1298:
1294:
1289:
1288:
1286:
1285:abelian group
1279:
1278:
1277:
1275:
1268:
1264:
1254:
1252:
1248:
1244:
1240:
1236:
1232:
1231:number theory
1228:
1224:
1220:
1216:
1212:
1207:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1175:
1170:
1165:
1161:
1156:
1152:
1148:
1143:
1141:
1137:
1133:
1129:
1125:
1121:
1116:
1114:
1113:
1108:
1104:
1100:
1096:
1092:
1091:abelian group
1088:
1084:
1080:
1068:
1063:
1061:
1056:
1054:
1049:
1048:
1046:
1045:
1037:
1034:
1032:
1029:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1003:
999:
996:
995:
991:
986:
985:
978:
977:
973:
972:
968:
965:
963:
960:
958:
955:
954:
949:
944:
943:
936:
935:
931:
929:
926:
925:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
892:
887:
882:
881:
876:
875:
868:
865:
863:
862:Division ring
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
819:
814:
809:
808:
803:
802:
795:
792:
790:
787:
785:
784:Abelian group
782:
781:
777:
774:
772:
769:
767:
763:
760:
758:
755:
754:
750:
745:
744:
741:
738:
737:
734:
732:
728:
724:
720:
716:
712:
708:
704:
700:
696:
692:
688:
684:
680:
676:
672:
668:
664:
652:
647:
645:
640:
638:
633:
632:
630:
629:
624:
623:
617:
613:
612:
611:
610:
609:
604:
603:
602:
597:
596:
595:
588:
584:
582:
578:
576:
572:
570:
569:Division ring
566:
565:
564:
563:
557:
552:
551:
523:
507:
505:
499:
483:
469:
468:-adic numbers
467:
462:
446:
432:
430:
425:
423:
419:
417:
410:
408:
404:
403:
402:
401:
400:
391:
387:
385:
381:
377:
373:
372:
371:
367:
365:
361:
359:
355:
353:
349:
347:
343:
341:
337:
336:
335:
331:
330:
329:
328:
322:
317:
316:
307:
303:
302:
301:
297:
293:
289:
287:
283:
282:
281:
277:
273:
269:
268:
267:
263:
262:
261:
260:
235:
231:
222:
219:
212:
211:Terminal ring
208:
185:
181:
180:
179:
175:
173:
169:
167:
163:
161:
157:
155:
151:
150:
149:
148:
147:
140:
136:
134:
130:
128:
124:
123:
122:
121:
120:
113:
109:
107:
103:
101:
97:
93:
89:
87:
83:
82:
81:
80:Quotient ring
77:
75:
71:
69:
65:
64:
63:
62:
53:
52:
49:
44:β Ring theory
43:
39:
38:
33:
19:
21620:
21601:(1): 18β35.
21598:
21592:
21560:
21554:
21539:
21521:
21513:
21452:
21448:
21432:
21428:
21400:
21396:
21393:Fraenkel, A.
21365:
21334:
21314:
21305:Local fields
21304:
21278:(1): 58β62,
21275:
21271:
21244:
21217:
21185:
21171:
21168:Knuth, D. E.
21136:
21130:
21104:
21092:
21073:
21069:
21045:
21031:
21022:
21018:
21000:
20982:
20964:
20955:
20936:
20909:
20887:
20855:
20846:
20825:
20787:
20765:
20743:
20721:
20687:
20664:
20652:
20648:
20636:
20632:
20610:
20591:
20569:
20550:
20533:
20515:, Springer,
20512:
20500:
20497:Bourbaki, N.
20488:
20485:Bourbaki, N.
20476:
20460:
20439:
20430:
20411:
20392:
20373:
20362:
20350:
20344:Serre (1950)
20339:
20327:
20311:
20306:, Ch XIV, Β§2
20299:
20284:
20272:
20265:Milne (2012)
20260:
20253:Milne (2012)
20248:
20236:
20224:
20212:
20185:
20173:
20161:
20156:, p. 98
20149:
20137:
20125:
20113:
20102:
20091:
20082:
20073:
20062:
20050:
20045:, p. 42
20038:
20026:
20021:, p. 11
20014:
20009:, p. 96
20002:
19995:Artin (2018)
19990:
19979:
19968:
19958:
19950:
19945:, p. 29
19925:
19917:
19905:
19893:
19878:
19867:
19862:, p. 27
19855:
19841:
19834:Serre (1979)
19829:
19822:Serre (2006)
19817:
19805:
19793:
19781:
19772:
19763:
19751:
19746:, p. 83
19731:, p. 85
19724:
19712:
19689:
19680:
19667:
19653:
19648:
19631:
19597:Regular ring
19592:Reduced ring
19587:Poisson ring
19582:Ordered ring
19538:Boolean ring
19531:
19511:Ring of sets
19456:
19452:
19396:
19379:Witt vectors
19368:
19363:
19318:
19303:over a base
19300:
19298:
18957:(addition),
18899:
18881:
18812:
18803:
18797:
18793:
18785:
18777:
18771:
18765:
18749:
18737:
18729:
18708:
18681:
18676:
18639:
18635:
18631:
18627:
18623:
18619:
18604:
18599:
18578:
18574:
18567:
18547:
18539:
18535:
18519:
18515:
18511:
18507:
18500:
18351:
18347:
18341:
18337:
18330:
18318:
18302:
18290:
18282:
18278:
18274:vector space
18216:
18210:
18180:
18144:
18122:Hopf algebra
18115:
18086:
18062:
18056:
18052:
18028:vector space
17958:
17844:
17828:
17582:
17578:
17574:
17537:
17533:
17523:
17519:
17515:
17451:
17409:vector space
17394:
17382:Novikov ring
17371:
17367:
17332:
17328:
17223:well ordered
17110:
17106:
17051:
17044:
17040:
17017:
17013:
17009:
17005:
17001:
16997:
16993:
16987:
16983:
16973:
16967:
16957:
16943:
16926:
16844:
16840:
16833:is split by
16787:
16783:
16775:
16771:
16726:
16720:through the
16571:
16567:
16556:
16547:Brauer group
16335:
16325:
16318:
16311:
16307:
16301:over a ring
16269:For a field
16268:
16243:
16239:
16188:
16176:
16172:
16165:
16161:
16082:is a finite
16050:
16047:
16030:Weyl algebra
16027:
15994:
15984:For a field
15972:
15967:
15930:
15924:
15922:
15896:
15889:
15886:
15877:
15863:
15851:
15845:
15839:
15833:
15827:
15821:
15815:
15809:
15803:
15798:
15797:
15791:
15782:
15775:
15750:
15746:
15581:
15530:
15448:
15438:
15434:
15430:
15415:
15411:number field
15381:is called a
15372:
15234:
15228:
15225:
15220:
15179:
15026:
15020:
15016:
14981:
14960:
14953:
14947:
14940:
14933:
14930:
14887:
14881:
14802:metric space
14293:
14263:
14081:
14067:
13929:
13331:
13226:
12897:
12689:localization
12688:
12438:localization
12435:
12432:Localization
12423:
12421:
12415:
12411:
12404:
12400:
12395:
12391:
12384:
12380:
12373:
12369:
12361:
12357:
12278:
12274:
12270:
12263:
12259:
12254:
12250:
12243:
12239:
12234:
12230:
12225:
12221:
12214:
12210:
12205:
12201:
12195:
12191:
12184:
12180:
12175:
12171:
12165:
12158:
12154:
12143:
12136:
12117:
11709:
11692:
11688:
11682:
11678:
11629:is the ring
11608:
11604:
11587:
11575:
11570:
11557:
11552:
11536:over it are
11531:
11526:
11517:
11506:
11265:
11260:
11244:
11237:
11232:
11227:
11222:
11214:
11210:
11207:
11086:
11081:
11076:
10922:
10915:
10910:
10877:
10872:
10851:
10821:
10818:
10700:
10622:
10616:
10598:
10591:
10541:
10477:
10473:
10450:
10282:
10267:
10263:
10257:
10253:
10249:
10245:
10241:
10237:
10221:
10217:
10213:
10209:
10205:
10199:
10192:
10188:
10184:
10173:
10167:
10161:
10155:
10152:
9994:
9976:
9930:
9926:
9916:
9912:
9908:
9902:
9898:
9892:
9888:
9884:substitution
9881:
9784:
9778:
9768:
9758:
9744:
9733:
9575:
9398:
9372:
9305:
9301:
9287:
8804:
8565:
8561:
8440:
8371:
8362:
8358:
8345:
8338:
8325:
8318:
8315:
8033:
8029:
8015:
7989:is called a
7972:
7968:
7958:
7954:
7950:
7946:
7932:
7928:
7924:
7921:
7910:
7906:
7902:
7890:
7887:
7880:
7876:
7872:
7868:
7861:
7857:
7853:
7849:
7836:
7833:left modules
7832:
7826:
7645:
7639:
7629:
7623:
7601:
7597:
7593:
7562:vector space
7557:
7555:
7534:
7530:
7471:
7467:
7461:
7457:
7453:
7450:
7354:
7350:
7344:
7340:
7336:
7326:
7322:
7318:
7312:
7308:
7304:
7300:
7297:
7292:
7287:
7283:
7281:. The ring
7273:
7269:
7266:
7100:
7096:
7092:
7086:
7074:
7070:
7055:
7051:
7039:
6997:
6967:
6963:
6959:
6956:
6949:is called a
6888:
6835:
6831:
6827:Galois group
6809:
6804:
6651:{0, 1, 2, 3}
6636:
6617:
6611:
6608:
6431:
6425:
6406:
6398:
6394:
6391:from a ring
6388:homomorphism
6386:
6384:
6375:Homomorphism
6367:
6363:
6357:
6353:
6347:
6343:
6337:
6331:
6205:is called a
6196:
6177:
6170:
6151:
6144:
6130:
6124:
6113:
6095:
6078:
6074:
6065:
6061:
6055:is a subset
6052:
6047:
6043:
6036:
5896:denotes the
5880:
5876:
5843:
5837:
5817:
5759:
5744:
5740:
5717:
5709:
5704:
5694:
5683:
5665:
5648:
5638:
5630:
5604:
5412:
5405:
5401:
5389:are in
5381:
5377:
5359:
5342:
5338:
5334:
5317:is called a
5308:
5283:
5268:
5264:
5257:
5251:
5241:
5226:
5216:
5212:
5186:
5182:nonzero ring
5171:
5164:
5153:
5146:
5126:zero divisor
5123:
5111:
5107:
5101:
5098:
5094:
5087:
5081:
5078:
5074:
5067:
5060:
5054:
5050:
5042:
5038:
5033:
5029:
5024:
5020:
5010:
4943:elements of
4882:
4599:
4589:
4583:
4565:
4547:
4530:
4525:
4513:over a ring
4501:
4494:
4484:
4470:
4456:
4454:itself. For
4445:
4408:Boolean ring
4404:intersection
4294:The ring of
4228:The ring of
4166:real-valued
4090:The algebra
4075:
4073:The algebra
4013:
3997:
3977:
3974:
3964:
3961:Emmy Noether
3946:
3935:
3929:
3919:
3916:
3629:
3625:
3613:
3604:Ernst Kummer
3589:
3552:
3534:
3177:
3032:
2806:
2681:
2483:The product
2367:
2363:
2357:
2353:
2343:
2339:
2329:
2325:
2142:
2124:
2120:
2065:
2061:
2054:(zero is an
2050:
2046:
2022:
2014:
1869:
1846:Illustration
1837:
1831:
1827:
1823:
1819:
1811:
1805:is called a
1796:
1791:
1787:
1781:
1776:
1765:
1758:
1727:
1716:
1710:
1706:
1699:
1686:
1682:
1678:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1634:
1630:
1626:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1589:distributive
1562:
1558:
1551:
1547:
1522:
1518:
1514:
1507:
1503:
1499:
1495:
1491:
1487:
1458:
1450:
1446:
1436:
1402:
1398:
1369:
1365:
1359:
1355:
1351:
1347:
1327:
1323:
1319:
1312:
1308:
1304:
1300:
1296:
1292:
1273:
1262:
1260:
1208:
1173:
1163:
1159:
1144:
1117:
1110:
1106:
1099:distributive
1086:
1082:
1078:
1077:Formally, a
1076:
1036:Hopf algebra
974:
967:Vector space
932:
872:
821:
812:
801:Group theory
799:
764: /
731:power series
686:
666:
660:
620:
606:
605:
601:Free algebra
599:
598:
592:
591:
560:
503:
465:
428:
397:
396:
376:Finite field
325:
272:Finite field
258:
257:
184:Initial ring
144:
143:
117:
116:
60:
59:
21644:Ring theory
21218:Local rings
21181:Korn, T. M.
21095:. Springer.
21076:: 795β799.
20784:Lang, Serge
20503:. Springer.
20334:, Ch IV, Β§2
20304:Lang (2002)
20289:Cohn (1995)
20229:Cohn (1995)
20217:Lang (2002)
20205:Cohn (2003)
20166:Cohn (2003)
20132:, p. 7
19883:Cohn (1980)
19824:, p. 3
19812:, Ch V, Β§3.
19810:Lang (2005)
19744:Lang (2002)
19558:Finite ring
19301:ring scheme
19295:Ring scheme
18900:ring object
18770:(sometimes
18756:Lie algebra
18690:-group (by
18311:, where if
17961:graded ring
17527:to be both
17470:is given a
17227:convolution
16632:local field
16283:simple ring
16159:where each
16051:For a ring
16034:simple ring
15874:quaternions
15399:prime ideal
14060:-modules.)
11511:states any
10866:matrix ring
10844:Matrix ring
10606:containing
10581:defined by
8569:. Then the
8356:. The ring
7985:, the ring
7843:instead of
7316:, given by
7293:factor ring
7091:; then the
6898:epimorphism
6818:called the
6619:isomorphism
6207:prime ideal
6160:along with
6069:. A subset
6053:right ideal
5806:centralizer
5639:For a ring
5617:containing
5602:, however.
5445:polynomials
5065:of a ring:
4753:convolution
4624:The set of
4575:free module
4511:left module
4478:, then the
4081:polynomials
3959:. In 1921,
3621:equivalence
3598:. In 1871,
3580:ring theory
3554:Matrix ring
2118:satisfying
1768:commutative
1383:commutative
1341:associative
1274:ring axioms
1180:group rings
1128:ring theory
1095:associative
1021:Lie algebra
1006:Associative
910:Total order
900:Semilattice
874:Ring theory
719:polynomials
679:commutative
663:mathematics
581:Simple ring
292:Jordan ring
166:Graded ring
48:Ring theory
18:Unital ring
21633:Categories
21204:Milne, J.
20818:0984.00001
20579:0824750330
20491:. Hermann.
20384:References
20318:, p.
19798:Lam (2001)
19573:Noetherian
19568:Local ring
19381:of length
19377:-isotypic
18813:Examples:
18698:being its
18530:associated
18227:under the
18205:See also:
18118:group ring
17529:continuous
17380:See also:
17343:group ring
17326:such that
17104:such that
17052:Examples:
17038:such that
16829:(that is,
16346:such that
16044:Properties
15353:See also:
15006:to a ring
14952:(see also
14083:completion
14064:Completion
13231:is a left
13048:local ring
13046:is then a
12367:such that
12199:such that
10830:(in fact,
10610:such that
10371:such that
9584:over
9383:adele ring
8850:be rings,
8380:with
7865:) becomes
7406:such that
7026:-module).
7002:to a ring
6637:Examples:
6403:to a ring
6077:or simply
6059:such that
5844:left ideal
5730:such that
5222:projection
5189:idempotent
5162:such that
5144:such that
5128:of a ring
4697:invertible
4561:group ring
4381:Hecke ring
4164:continuous
4034:See also:
3955:to have a
3565:See also:
2843:0, 1, 2, 3
2137:See also:
2095:If a ring
2076:in a ring
2044:, one has
2040:in a ring
1574:(that is,
1545:such that
1530:(that is,
1455:(that is,
1444:such that
1414:(that is,
1396:such that
1377:(that is,
1335:(that is,
1257:Definition
1153:, and the
587:Commutator
346:GCD domain
21544:MIT Press
21469:121594471
21417:118962421
21285:1404.0135
21216:(1962) ,
21189:. Dover.
21153:0003-486X
20939:. Dover.
20590:(1994) .
19705:Citations
19460:from the
19435:→
19429:∧
19420:μ
19336:
19258:forgetful
19252:⟶
19228:→
19184:→
19159:−
19153:
19100:→
19060:→
19019:→
18979:→
18968:×
18932:→
18921:×
18846:…
18720:morphisms
18440:⋅
17912:∞
17897:⨁
17871:∗
17744:λ
17674:−
17667:λ
17648:λ
17633:∑
17605:λ
17557:, or the
17514:β
:
17497:→
17491:×
17301:−
17272:∈
17265:∑
17243:∗
16991:nonzero,
16950:valuation
16903:∗
16875:
16804:⊗
16747:⊗
16743:−
16681:
16599:
16561:. By the
16519:⊗
16453:≃
16434:⊗
16384:⊗
16377:≈
16358:⊗
16213:⊗
16199:separable
16131:
16094:∏
15697:λ
15693:−
15542:⋅
15492:…
15330:⊗
15312:⊗
15297:⊗
15258:⊗
15246:-modules
15215:from the
15191:↦
15136:∈
15124:∣
15115:−
14988:free ring
14535:−
14499:given by
14422:↦
14411:is a map
14219:^
14210:→
14127:
14122:←
14108:^
14039:−
13994:−
13922:is exact.
13907:→
13896:→
13890:→
13879:→
13826:→
13808:→
13793:→
13775:→
13742:→
13731:→
13725:→
13714:→
13703:whenever
13682:−
13647:→
13635:−
13615:→
13603:−
13588:→
13576:−
13556:→
13511:−
13496:
13491:→
13471:−
13426:−
13377:−
13358:↦
13306:⊗
13293:−
13266:−
13050:with the
12994:−
12923:−
12874:−
12833:−
12798:≥
12785:∈
12731:−
12666:−
12646:The ring
12622:−
12570:−
12521:−
12506:→
12475:−
12331:∏
12300:
12295:←
12268:whenever
12091:
12086:→
11982:−
11974:
11969:→
11889:
11884:→
11865:¯
11808:⋯
11773:
11768:→
11754:⋯
11647:
11642:→
11470:
11454:
11417:∏
11413:≃
11401:
11350:-modules
11315:⊕
11285:⨁
11171:↦
11152:
11139:→
11119:⊕
11112:
11033:⊕
11029:→
11012:⊕
10973:
10960:≃
10941:
10801:∈
10762:∞
10753:∑
10660:…
10563:¯
10525:¯
10517:↦
10504:→
10426:¯
10423:ϕ
10384:¯
10381:ϕ
10356:→
10336:¯
10333:ϕ
10300:→
10291:ϕ
10165:and
10099:↦
10077:→
9995:Example:
9988:and
9948:↦
9855:↦
9842:→
9796:⊆
9668:…
9617:…
9553:∈
9534:≥
9528:∣
9496:⋯
9485:−
9469:−
9215:∈
9185:⋯
9120:⊆
9091:≠
9030:⊕
9027:⋯
9024:⊕
8877:∏
8868:→
8835:≤
8829:≤
8760:…
8729:↦
8692:⋂
8634:∏
8630:≃
8592:⋂
8559:whenever
8470:⋯
8411:∈
8404:∏
8284:⋅
8258:⋅
8210:⋅
7590:operation
7513:→
7507:
7488:¯
7430:∘
7425:¯
7391:→
7372:¯
6915:→
6869:↦
6766:−
6752:↦
6740:→
6632:bijective
6541:∗
6517:⋅
6488:‡
6307:∈
6281:∈
6255:∈
6226:∈
6136:principal
6016:∈
5991:∈
5938:⋯
5900:-span of
5659:and
5309:A subset
5235:having a
5169:for some
4969:∏
4911:…
4829:−
4803:∞
4798:∞
4795:−
4791:∫
4769:∗
4619:Non-rings
4392:power set
4315:^
4198:pointwise
4168:functions
4120:…
3898:⋮
3887:⋮
3874:−
3849:−
3810:−
3767:−
3745:−
3715:−
3672:−
3200:
3152:∈
3065:
3009:¯
2996:¯
2989:−
2978:¯
2970:−
2945:¯
2938:−
2927:¯
2919:−
2858:¯
2824:¯
2662:¯
2649:¯
2641:⋅
2636:¯
2609:¯
2596:¯
2588:⋅
2583:¯
2509:¯
2501:⋅
2496:¯
2464:¯
2451:¯
2438:¯
2411:¯
2398:¯
2385:¯
2270:¯
2257:¯
2221:¯
2208:¯
2195:¯
2182:¯
2090:zero ring
2049:0 = 0 = 0
2000:…
1955:−
1946:−
1937:−
1928:−
1919:−
1913:…
1425:For each
1233:, and of
1031:Bialgebra
837:Near-ring
794:Lie group
762:Semigroup
727:functions
528:∞
306:Semifield
21619:(1985),
21512:(1996),
21443:(1921).
21364:(1975).
21303:(1979),
21294:48666015
21266:(2019),
21183:(2000).
21170:(1998).
21103:(1994).
21042:(1995),
20958:. Wiley.
20907:(1930),
20881:. v2.23.
20786:(2002),
20764:(2003).
20742:(2001).
20720:(1999).
20685:(1974),
20663:(2009).
20532:(1995).
20511:(2003),
20499:(1989).
20487:(1964).
20475:(1969).
20459:(2018).
19607:SBI ring
19563:Lie ring
19516:Semiring
19479:See also
19407:spectrum
18888:products
18802:for all
18768:semiring
18762:Semiring
18706:-group.
18255:-modules
17472:topology
17004:) β₯ min{
16768:induces
16727:Now, if
16070:artinian
15939:Examples
15660:Now, if
15575:we make
15531:Letting
15433: :
15014:so that
13903:″
13886:′
13822:″
13789:′
13738:″
13721:′
13622:″
13563:′
12378:maps to
11614:for all
11542:category
11218:induces
10886:-module
10832:complete
10550:) where
10546:maps to
10276:at
9815:of
8352:in
8316:for all
7927: :
7637:and all
7580:-module
7564:(over a
7456: :
7339: :
7303: :
7267:for all
7018:to
6979:of
6962: :
6419:to
5820:in
5780:of
5772:of
5752:in
5748:for any
5688:of
5330:restrict
5105:for all
5092:. Then
5018:and let
4742:semiring
4499:of
4437:matrices
3586:Dedekind
2244:The sum
1860:addition
1856:integers
1733:" (IPA:
1676:for all
1624:for all
1566:for all
1512:for all
1406:for all
1363:for all
1317:for all
1251:analysis
1247:geometry
1219:Fraenkel
1211:Dedekind
1204:topology
1083:addition
867:Lie ring
832:Semiring
711:integers
707:integers
699:addition
300:Semiring
286:Lie ring
68:Subrings
21577:2974935
21236:0155856
21161:1969205
20929:0009016
20847:Algebra
20810:1878556
20788:Algebra
20711:0345945
20542:1322960
20461:Algebra
20293:pg. 242
19353:
19325:
18800:β
0 = 0
18295:be its
18266:-module
18253:
18231:
18116:To any
18087:To any
18046:and an
18012:spheres
17845:To any
17733:
17711:
17458:A ring
17447:
17425:
17123:is the
16774:) β Br(
16666:
16637:
16336:similar
15375:nonzero
15369:Domains
15086:
15061:
14927:
14895:
14871:
14842:
14838:
14806:
14798:
14776:
14772:
14750:
14497:
14475:
14471:
14449:
14409:
14387:
14379:
14354:
14343:
14311:
14288:
14266:
13960:functor
12282:. Then
11518:A ring
10577:is the
10182:. Then
9972:
9937:
9906:, then
9742:, then
8395:, then
8026:product
7995:algebra
7079:, view
7012:algebra
6932:
6902:
6884:
6856:
6802:. Then
6729:, then
6716:
6681:
6677:
6655:
6409:, β‘, β)
6122:, then
6093:, then
5888:are in
5818:central
5796:. Then
5756:. Then
5596:
5571:
5567:
5542:
5534:
5509:
5505:
5483:
5479:
5448:
5437:
5415:
5346:making
5319:subring
5305:Subring
5299:Subring
5124:A left
4734:
4709:
4703:to get
4689:
4655:
4651:
4629:
4581:having
4370:
4341:
4333:
4299:
4285:
4260:
4256:
4234:
4194:
4172:
4068:-module
3632:+ 1 = 0
3610:Hilbert
3561:History
2909:
2874:
2802:
2764:
2748:
2723:
2719:
2684:
2560:
2525:
2321:
2286:
1901:numbers
1897:
1872:
1840:algebra
1578:is the
1462:is the
1418:is the
1223:Noether
1215:Hilbert
998:Algebra
990:Algebra
895:Lattice
886:Lattice
502:PrΓΌfer
104:β’
21575:
21531:
21492:
21467:
21415:
21372:
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21349:(also
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20728:
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20699:
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20576:
20557:
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20418:
20399:
19305:scheme
18505:as in
18501:where
18327:is in
18323:, and
18315:is in
18219:, the
18213:monoid
18181:Every
17566:Ξ»-ring
17176:, let
16584:; cf.
16415:Since
15979:simple
15950:simple
15878:finite
15847:fields
15383:domain
15279:is an
14885:|
14879:|
14568:where
14080:. The
12389:under
12148:(or a
11540:: the
11059:is an
9886:). If
9738:is an
8368:(1, 1)
7979:center
7619:axioms
7584:is an
7546:Module
7533:/ ker
7470:= ker
7106:cosets
6977:kernel
6173:simple
6118:is in
5784:, let
5770:center
5714:, let
5600:subrng
5385:, and
5174:> 0
4474:is an
4459:> 1
4051:fields
4004:Poonen
3933:, and
3890:
3634:then:
3465:while
3128:
1790:means
1550:Β· 1 =
1480:monoid
1401:+ 0 =
1283:is an
1221:, and
1198:, and
1149:of an
1140:fields
1109:. See
1026:Graded
957:Module
948:Module
847:Domain
766:Monoid
729:, and
675:fields
154:Module
127:Kernel
21573:JSTOR
21465:S2CID
21413:S2CID
21290:JSTOR
21280:arXiv
21157:JSTOR
20207:, 4.4
19887:p. 49
19623:Notes
19405:is a
19385:over
19323:over
18538:, +,
18089:group
17462:is a
17047:) β₯ 0
16981:with
16670:then
16247:. If
16008:order
15870:field
15799:rings
15409:in a
15169:over
14986:be a
14877:with
14800:as a
14447:from
14253:, by
13958:is a
13868:then
12127:stalk
11622:) of
11380:then
11230:) β M
11079:= End
10828:local
10256:)) /
9752:. If
7962:. If
7898:basis
7856:) = (
7576:, an
7566:field
7357:) = 0
7073:, +,
7062:ideal
7054:, +,
7014:over
6415:from
6397:, +,
6180:chain
6085:. If
6079:ideal
6037:then
5892:. If
5828:Ideal
5441:field
5247:group
4857:Then
4693:group
4577:over
4573:is a
4569:over
4553:group
4551:is a
4509:is a
4461:(and
3401:then
3039:field
2811:" or
2809:mod 4
2682:Then
2074:0 = 1
1807:field
1666:) + (
1614:) + (
1606:) = (
1478:is a
1453:) = 0
1265:is a
1171:with
1167:real
1097:, is
1085:and
992:-like
950:-like
888:-like
857:Field
815:-like
789:Magma
757:Group
751:-like
749:Group
689:is a
667:rings
506:-ring
370:Field
266:Field
74:Ideal
61:Rings
21529:ISBN
21490:ISBN
21401:1915
21370:ISBN
21340:ISBN
21320:ISBN
21250:ISBN
21222:ISBN
21191:ISBN
21149:ISSN
21113:ISBN
21052:ISBN
21005:ISBN
20987:ISBN
20941:ISBN
20915:ISBN
20892:ISBN
20864:ISBN
20830:ISBN
20796:ISBN
20770:ISBN
20748:ISBN
20726:ISBN
20697:ISBN
20669:ISBN
20615:ISBN
20596:ISBN
20574:ISBN
20555:ISBN
20517:ISBN
20416:ISBN
20397:ISBN
19614:and
19575:and
19401:, a
19333:Spec
18882:Let
18864:the
18792:0 β
18788:, +)
18780:, +)
18630:) =
18581:, +)
18570:, +)
18550:, +)
18514:) +
18345:and
18329:End(
18317:End(
18301:End(
18289:End(
18285:, +)
18279:Let
18016:tori
18014:and
17799:the
17549:-by-
17419:-by-
17384:and
17022:The
16342:and
16330:and
16319:The
16273:, a
16072:and
16028:The
15793:rngs
15428:and
15357:and
15226:Let
14655:and
14381:The
14068:Let
14015:and
13934:, a
12974:for
12436:The
12065:The
11841:The
11588:Let
11507:The
10852:Let
10846:and
10623:Let
10413:and
10248:) β
10216:) β
10203:and
9915:) =
9896:and
9785:Let
8996:and
8441:Let
8336:and
8020:and
8016:Let
7905:(β1)
7089:, +)
6825:The
6794:Let
6128:and
5975:that
5967:such
5884:and
5866:and
5860:x, y
5838:Let
5365:x, y
5358:1 β
5229:unit
5085:for
5072:and
5049:1 β€
5047:for
4747:Let
4606:The
4555:and
4538:The
4493:End(
4432:-by-
4379:The
4203:Let
3337:and
2623:and
2425:and
2337:(as
2114:and
2106:The
2060:(β1)
1862:and
1854:The
1799:zero
1788:ring
1654:) Β·
1557:1 Β·
1555:and
1494:) Β·
1449:+ (β
1299:) +
1263:ring
1249:and
1241:and
1134:and
1079:ring
822:Ring
813:Ring
701:and
687:ring
681:and
669:are
21603:doi
21565:doi
21561:103
21457:doi
21405:doi
21141:doi
21109:AMS
21078:doi
20814:Zbl
19655:rng
19397:In
19373:of
19150:Hom
18902:in
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18806:in
18772:rig
18740:rng
18734:Rng
18674:End
18672:to
18648:in
18613:of
18597:End
18587:of
18577:= (
18556:in
18215:in
18120:or
18097:act
18067:of
17590:th
17403:An
17345:of
17217:to
17026:of
17020:)}.
17012:),
16977:in
16944:If
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16839:Br(
16782:Br(
16770:Br(
16588:).
16570:Br(
16555:Br(
16549:of
16255:.)
16068:is
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15177:.)
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11765:lim
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11611:+ 1
11461:End
11392:End
11258:End
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11074:in
10932:End
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10834:).
10698:.)
10596:in
10467:in
10319:in
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10261:is
10234:in
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9297:= 0
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8740:mod
8686:mod
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8332:in
7981:of
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7653:,
7649:in
7633:in
7504:ker
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7067:of
6896:An
6891:β 0
6787:of
6721:If
6435:in
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6296:or
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6114:If
6000:and
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5870:in
5862:in
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5726:in
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5187:An
5167:= 0
5149:= 0
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5070:= 1
5016:= 1
4939:of
4563:of
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4523:End
4482:of
4468:If
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2284:in
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1731:rng
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19736:^
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7875:=
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13483:=
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9007:=
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8874:=
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8732:(
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8119:(
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7867:(
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7860:)
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7850:x
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7799:(
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7793:=
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7778:(
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7765:=
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7436:.
7433:p
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7417:=
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7218:+
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7212:(
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