3126:. So all properties of localizations can be deduced from the universal property, independently from the way they have been constructed. Moreover, many important properties of localization are easily deduced from the general properties of universal properties, while their direct proof may be together technical, straightforward and boring.
6802:
7245:
6954:
7099:
6681:
3629:
2591:
8798:
5057:
2717:
9384:
8931:
9044:
6687:
8417:
1700:
8089:
7105:
3634:
This may seem a rather tricky way of expressing the universal property, but it is useful for showing easily many properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor.
8640:
7397:
6814:
3431:
1089:
4895:
9452:, which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see
9424:
Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because of
5266:
7931:
1156:
2179:
7995:
9501:
7498:
6960:
7314:
4408:
Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerning
2347:
1859:
3717:
8170:
7453:
1282:
9155:
6551:
4664:
4254:
774:
2280:
1960:
567:
9835:
One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse
8264:
6590:
4059:
6597:
2830:
675:
7351:
3474:
2407:
1409:
503:
8954:
6181:
3265:
3172:
3083:
3828:
9659:
9593:
9420:
9248:
5610:
4760:
4351:
870:
9277:
5925:
4949:
2451:
9315:
6391:
4385:
2789:
2754:
459:
9683:
9617:
9179:
7271:
6483:
Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation. In what follows,
5896:
5868:
5424:
4973:
4692:
4278:
4207:
2041:
1556:
979:
805:
85:
9782:
8296:
7606:
3670:
3367:
3343:
3208:
8675:
6087:
6113:
6016:
5312:
5131:
4988:
4565:
4513:
3972:
3772:
1498:
160:
134:
7755:
4802:
4176:
3300:
5497:
5163:
3466:
7821:
4456:
1781:
7853:
6479:
6298:
5533:
5468:
5392:
4728:
2893:
2107:
1740:
1322:
1225:
618:
9216:
9105:
8509:
8472:
7787:
7673:
6258:
6225:
5695:
5662:
5566:
5341:
4613:
4315:
4092:
3921:
3861:
3750:
3120:
3022:
2966:
2933:
2602:
2443:
1995:
1618:
1589:
1466:
1370:
1189:
1008:
931:
838:
708:
327:
252:
9320:
7644:
6447:
6421:
5974:
5725:
2205:
1885:
738:
1527:
9076:
2071:
8824:
4178:(The proof consists of showing that this ring satisfies the above universal property.) This sort of localization plays a fundamental role in the definition of an
8966:
8094:
It is straightforward to check that these operations are well-defined, that is, they give the same result for different choices of representatives of fractions.
6797:{\displaystyle I\subseteq J\quad \ \implies \quad \ S^{-1}I\subseteq S^{-1}J\quad \ {\text{and}}\quad \ \operatorname {sat} (I)\subseteq \operatorname {sat} (J)}
8663:
5183:
3995:
2857:
2300:
8311:
9828:
of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved is the
8526:
make that localization preserves many properties of modules and rings, and is compatible with solutions of other universal properties. For example, the
1623:
7240:{\displaystyle S^{-1}(I\cdot J)=S^{-1}I\cdot S^{-1}J,\qquad \quad \operatorname {sat} (I\cdot J)=\operatorname {sat} (I)\cdot \operatorname {sat} (J)}
8019:
9429:. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring is
8535:
6949:{\displaystyle S^{-1}(I\cap J)=S^{-1}I\cap S^{-1}J,\qquad \,\operatorname {sat} (I\cap J)=\operatorname {sat} (I)\cap \operatorname {sat} (J)}
3389:
of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let
7356:
3392:
1013:
4824:
5191:
7861:
1094:
5980:. Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a
2112:
422:. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.
7937:
573:. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element".
17:
5828:
by a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed as
9465:
7458:
10100:
10071:
9977:
7094:{\displaystyle S^{-1}(I+J)=S^{-1}I+S^{-1}J,\qquad \operatorname {sat} (I+J)=\operatorname {sat} (I)+\operatorname {sat} (J)}
5734:
Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must know
10161:
9878:
402:
must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to
410:
that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of
7280:
2305:
10037:
10010:
9994:
1786:
3679:
8107:
7402:
1230:
9124:
6514:
4633:
4391:, because many properties of a commutative ring can be read on its local rings. Such a property is often called a
4223:
743:
2210:
1890:
508:
6676:{\displaystyle 1\in S^{-1}I\quad \iff \quad 1\in \operatorname {sat} (I)\quad \iff \quad S\cap I\neq \emptyset }
8214:
6560:
4004:
3624:{\displaystyle \hom _{\mathcal {C}}((R,S),{\mathcal {F}}(T,U))\to \hom _{\mathcal {D}}((S^{-1}R,j(S)),(T,U)).}
5765:, that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of
2802:
623:
8960:, that is the ideal of the elements of the ring that map to zero all elements of the module. In particular,
7319:
5798:
2352:
1375:
1339:
are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case,
464:
8939:
6141:
3433:
be the forgetful functor that forgets that the elements of the second element of the pair are invertible.
3225:
3132:
3043:
2586:{\displaystyle {\frac {r_{1}}{s_{1}}}+{\frac {r_{2}}{s_{2}}}={\frac {r_{1}s_{2}+r_{2}s_{1}}{s_{1}s_{2}}},}
3792:
365:
290:
9632:
9566:
9393:
9221:
5571:
4733:
4324:
843:
9873:
9253:
5901:
4925:
9293:
8793:{\displaystyle S^{-1}\operatorname {Hom} _{R}(M,N)\to \operatorname {Hom} _{S^{-1}R}(S^{-1}M,S^{-1}N)}
6341:
4363:
2762:
2727:
432:
9664:
9598:
9160:
8957:
8811:
8666:
8098:
7252:
5932:
5877:
5849:
5397:
5052:{\displaystyle R=\bigcap _{\mathfrak {p}}R_{\mathfrak {p}}=\bigcap _{\mathfrak {m}}R_{\mathfrak {m}}}
4954:
4780:
Localization commutes with formations of finite sums, products, intersections and radicals; e.g., if
4673:
4259:
4188:
2016:
1532:
955:
786:
60:
9755:
8269:
7584:
3647:
3348:
3324:
3181:
6067:
6096:
5999:
5282:
5110:
4531:
4479:
3938:
3755:
1471:
143:
117:
9519:
7760:
Addition and scalar multiplication are defined as for usual fractions (in the following formula,
7713:
5840:
4783:
4667:
4101:
3924:
3270:
3025:
2757:
777:
5473:
5139:
3439:
2712:{\displaystyle {\frac {r_{1}}{s_{1}}}\,{\frac {r_{2}}{s_{2}}}={\frac {r_{1}r_{2}}{s_{1}s_{2}}},}
10156:
9507:
9379:{\displaystyle {\mathfrak {p}}\,R_{\mathfrak {p}}={\mathfrak {p}}\otimes _{R}R_{\mathfrak {p}}}
8527:
7791:
5616:, that is, there is a unique isomorphism between them that fixes the images of the elements of
5427:
4422:
2794:
1760:
190:
7826:
6452:
6267:
5502:
5437:
5365:
4697:
2862:
2076:
1709:
1291:
1194:
587:
9844:
9188:
9081:
8481:
8444:
8176:
7763:
7649:
6230:
6197:
5667:
5634:
5613:
5538:
5317:
5063:
where the first intersection is over all prime ideals and the second over the maximal ideals.
4585:
4287:
4095:
4064:
3893:
3833:
3722:
3092:
2994:
2938:
2909:
2415:
1967:
1594:
1561:
1442:
1342:
1161:
984:
903:
810:
680:
388:
299:
224:
8926:{\displaystyle S^{-1}(\operatorname {Ann} _{R}(M))=\operatorname {Ann} _{S^{-1}R}(S^{-1}M),}
7623:
6426:
6400:
5949:
5704:
2184:
1864:
875:
In the remainder of this article, only localizations by a multiplicative set are considered.
713:
10151:
10110:
10081:
10047:
10020:
9821:
9039:{\displaystyle S^{-1}M=0\quad \iff \quad S\cap \operatorname {Ann} _{R}(M)\neq \emptyset ,}
7537:
5782:
4413:
1754:
1503:
47:
9052:
2045:
8:
9848:
9801:
9449:
8426:
8302:
6184:
5774:
4805:
4409:
4388:
1227:
This results from the defining property of a multiplicative set, which implies also that
174:
31:
10032:. Vol. 3 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xii+474.
10005:. Vol. 2 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xvi+428.
9743:
9711:
9434:
8648:
8523:
8412:{\displaystyle S^{-1}R\otimes _{R}f:\quad S^{-1}R\otimes _{R}M\to S^{-1}R\otimes _{R}N}
8179:) can be done by showing that the two definitions satisfy the same universal property.
7521:
5993:
5981:
5818:
5778:
5770:
5738:
5168:
3980:
3864:
3215:
3086:
2896:
2842:
2285:
934:
345:
263:
182:
170:
111:
55:
43:
35:
9796:
On the other hand, some properties are not local properties. For example, an infinite
414:. However, the same localization is obtained by using the multiplicatively closed set
10096:
10095:. Lecture Notes in Mathematics, Vol. 237. Berlin: Springer-Verlag. pp. vii+136.
10067:
10033:
10006:
9990:
9973:
9731:
7578:
5810:
4572:
4474:
3318:
2836:
2722:
1998:
194:
9965:
9812:, while all its local rings are fields, and therefore Noetherian integral domains.
7513:
5814:
5806:
3879:
3123:
1336:
349:
283:
5697:
are isomorphic if and only if they have the same saturation, or, equivalently, if
10131:
10106:
10077:
10063:
10043:
10016:
9809:
9805:
5976:
and its spectrum is the
Zariski open set of the prime ideals that do not contain
5790:
3786:
3775:
3306:
1695:{\displaystyle {\tfrac {a}{1}}={\tfrac {as}{s}}={\tfrac {0}{s}}={\tfrac {0}{1}}.}
889:
341:
137:
107:
10055:
9961:
9868:
9856:
9797:
9445:
9426:
9119:
8434:
8084:{\displaystyle r\,{\frac {m}{s}}={\frac {r}{1}}{\frac {m}{s}}={\frac {rm}{s}}.}
4458:
4392:
4218:
3382:
6064:" means that the property is considered after localization at the prime ideal
10145:
9829:
9511:
9440:
Properties of a ring that can be characterized on its local rings are called
9387:
8475:
6806:
6116:
5989:
5794:
5786:
4767:
4179:
6048:" means that the property is considered after localization by the powers of
9986:
9430:
6057:
4906:
4522:
4396:
3887:
3370:
3314:
2984:
1428:
1329:
353:
8635:{\displaystyle S^{-1}(M\otimes _{R}N)\to S^{-1}M\otimes _{S^{-1}R}S^{-1}N}
50:. That is, it introduces a new ring/module out of an existing ring/module
9721:
9115:
8512:
7274:
5844:
5802:
4616:
4568:
4210:
1324:
It is shown below that this is no longer true in general, typically when
781:
88:
1742:
The construction that follows is designed for taking this into account.
10117:
9280:
5928:
4763:
4576:
4318:
4284:
is a multiplicative set (by the definition of a prime ideal). The ring
271:
5801:). This correspondence has been generalized for making the set of the
10135:
8299:
8192:
7392:{\displaystyle {\mathfrak {p}}=\operatorname {sat} ({\mathfrak {p}})}
5793:
in such a way that the points of the algebraic set correspond to the
5069:
3426:{\displaystyle {\mathcal {F}}\colon {\mathcal {D}}\to {\mathcal {C}}}
3374:
2969:
1084:{\displaystyle {\tfrac {a}{s}}+{\tfrac {b}{t}}={\tfrac {at+bs}{st}},}
9459:
Many local properties are a consequence of the fact that the module
4890:{\displaystyle {\sqrt {I}}\cdot S^{-1}R={\sqrt {I\cdot S^{-1}R}}\,.}
5766:
5758:
5754:
5261:{\displaystyle {\hat {S}}=\{r\in R\colon \exists s\in R,rs\in S\}.}
4416:, or several multiplicative sets are considered in other sections.
3386:
10062:, Graduate Texts in Mathematics, vol. 150, Berlin, New York:
8097:
The localization of a module can be equivalently defined by using
7926:{\displaystyle {\frac {m}{s}}+{\frac {n}{t}}={\frac {tm+sn}{st}},}
3974:
is injective. The preceding example is a special case of this one.
2409:
is nonzero even though the fractions should be regarded as equal.
1151:{\displaystyle {\tfrac {a}{s}}\,{\tfrac {b}{t}}={\tfrac {ab}{st}}}
6019:
3673:
3310:
945:
163:
9989:. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag.
9510:
when the direct sum is taken over all prime ideals (or over all
620:
but other notations are commonly used in some special cases: if
9824:
is more difficult. While the localization exists for every set
5753:
originates in the general trend of modern mathematics to study
3378:
2174:{\displaystyle {\tfrac {r_{1}}{s_{1}}}={\tfrac {r_{2}}{s_{2}}}}
352:
as the field of fractions of the integers. For rings that have
7990:{\displaystyle {\frac {r}{s}}{\frac {m}{t}}={\frac {rm}{st}}.}
10120:, "Algebraic Number Theory," Springer, 2000. pages 3–4.
5898:", or "localization at a point". The resulting ring, denoted
5701:
belongs to one of the multiplicative sets, then there exists
3935:
is the largest multiplicative set such that the homomorphism
344:
the construction generalizes and follows closely that of the
201:, and one wants to study this variety "locally" near a point
5938:
The multiplicative set consists of all powers of an element
262:, and excludes information that is not "local", such as the
9847:. There is now a large mathematical theory about it, named
9496:{\displaystyle \bigoplus _{\mathfrak {p}}R_{\mathfrak {p}}}
7569:-module that is constructed exactly as the localization of
5835:
Two classes of localizations are more commonly considered:
505:
So, the denominators will belong to the multiplicative set
42:
is a formal way to introduce the "denominators" to a given
6093:
is prime, the nonzero prime ideals of the localization of
6089:. This terminology can be explained by the fact that, if
3309:, this can be expressed by saying that localization is a
9185:
is a multiplicative set. In this case, the localization
7573:, except that the numerators of the fractions belong to
7493:{\displaystyle \operatorname {sat} ({\mathfrak {p}})=R.}
6125:
1372:
consists of the rational numbers that can be written as
9444:, and are often the algebraic counterpart of geometric
5744:
9453:
8522:
This flatness and the fact that localization solves a
5087:. This bijection is induced by the given homomorphism
4909:
if and only if its total ring of fractions is reduced.
3871:. The preceding example is a special case of this one.
2767:
2732:
2325:
2310:
2146:
2117:
1678:
1663:
1643:
1628:
1537:
1380:
1241:
1127:
1112:
1099:
1048:
1033:
1018:
960:
469:
437:
356:, the construction is similar but requires more care.
169:
The technique has become fundamental, particularly in
114:: this case generalizes the construction of the field
9758:
9667:
9635:
9601:
9569:
9468:
9396:
9323:
9296:
9256:
9224:
9191:
9163:
9127:
9084:
9055:
8969:
8942:
8827:
8678:
8651:
8538:
8484:
8447:
8314:
8272:
8217:
8110:
8022:
7940:
7864:
7829:
7794:
7766:
7716:
7652:
7626:
7587:
7461:
7405:
7359:
7322:
7283:
7255:
7108:
6963:
6817:
6690:
6600:
6563:
6517:
6455:
6429:
6403:
6344:
6270:
6233:
6200:
6144:
6099:
6070:
6041:, depending on the localization that is considered. "
6002:
5952:
5904:
5880:
5852:
5707:
5670:
5637:
5574:
5541:
5505:
5476:
5440:
5400:
5368:
5320:
5285:
5194:
5171:
5142:
5113:
5102:
4991:
4957:
4928:
4827:
4786:
4736:
4700:
4676:
4636:
4588:
4534:
4482:
4425:
4366:
4327:
4290:
4262:
4226:
4191:
4104:
4067:
4007:
3983:
3941:
3896:
3836:
3795:
3758:
3725:
3682:
3650:
3477:
3442:
3395:
3351:
3327:
3273:
3228:
3184:
3135:
3095:
3046:
2997:
2941:
2912:
2865:
2845:
2805:
2765:
2730:
2605:
2454:
2418:
2355:
2308:
2288:
2213:
2187:
2115:
2079:
2048:
2019:
1970:
1893:
1867:
1789:
1763:
1712:
1626:
1597:
1564:
1535:
1506:
1474:
1445:
1378:
1345:
1294:
1233:
1197:
1164:
1097:
1016:
987:
958:
941:. As such, the localization of a domain is a domain.
906:
846:
813:
789:
746:
716:
683:
626:
590:
511:
467:
435:
302:
227:
146:
120:
63:
6397:, which can also defined as the set of the elements
5874:. In this case, one speaks of the "localization at
10087:
Matsumura. Commutative
Algebra. Benjamin-Cummings
2991:(that is the elements that are not zero divisors),
9776:
9677:
9653:
9611:
9587:
9495:
9414:
9378:
9309:
9271:
9242:
9210:
9173:
9149:
9099:
9070:
9038:
8948:
8925:
8792:
8657:
8634:
8503:
8466:
8411:
8290:
8258:
8164:
8083:
7989:
7925:
7847:
7815:
7781:
7749:
7667:
7638:
7600:
7492:
7447:
7391:
7345:
7308:
7265:
7239:
7093:
6948:
6796:
6675:
6584:
6545:
6473:
6441:
6415:
6385:
6292:
6252:
6219:
6175:
6107:
6081:
6022:, one refers to a property relative to an integer
6010:
5968:
5919:
5890:
5862:
5719:
5689:
5656:
5604:
5560:
5527:
5491:
5462:
5418:
5386:
5335:
5306:
5260:
5177:
5157:
5125:
5051:
4967:
4943:
4918:be an integral domain with the field of fractions
4889:
4796:
4754:
4722:
4686:
4658:
4607:
4559:
4507:
4450:
4379:
4345:
4309:
4272:
4248:
4201:
4170:
4086:
4053:
3989:
3966:
3915:
3855:
3822:
3766:
3744:
3711:
3664:
3623:
3460:
3425:
3361:
3337:
3294:
3259:
3210:is a ring homomorphism that maps every element of
3202:
3166:
3114:
3077:
3016:
2960:
2927:
2887:
2851:
2824:
2783:
2748:
2711:
2585:
2437:
2401:
2341:
2294:
2274:
2199:
2173:
2101:
2065:
2035:
1989:
1954:
1879:
1853:
1775:
1734:
1694:
1612:
1583:
1550:
1521:
1492:
1460:
1403:
1364:
1316:
1276:
1219:
1183:
1150:
1083:
1002:
973:
925:
864:
832:
799:
768:
732:
702:
669:
612:
561:
497:
453:
425:For example, the localization by a single element
321:
246:
154:
128:
79:
7309:{\displaystyle {\mathfrak {p}}\cap S=\emptyset ,}
10143:
5832:) that do not intersect the multiplicative set.
2342:{\displaystyle {\tfrac {a}{1}}={\tfrac {0}{1}},}
364:Localization is commonly done with respect to a
329:whose elements are fractions with numerators in
9960:
9939:
9897:
9851:, connecting with numerous other branches. The
9843:. This is done in many contexts in methods for
9525:
6122:or its complement in the set of prime numbers.
4399:if and only if all its local rings are regular.
1854:{\displaystyle (r_{1},s_{1})\sim (r_{2},s_{2})}
6134:be a multiplicative set in a commutative ring
5984:consisting of Zariski open sets of this form).
3712:{\displaystyle S=\mathbb {Z} \setminus \{0\},}
3468:of the universal property defines a bijection
1435:be a multiplicative set in a commutative ring
461:but also products of such fractions, such as
8474:-modules. In other words, localization is an
8165:{\displaystyle S^{-1}M=S^{-1}R\otimes _{R}M.}
7448:{\displaystyle S^{-1}{\mathfrak {p}}=S^{-1}R}
6183:be the canonical ring homomorphism. Given an
4624:
1277:{\displaystyle 1={\tfrac {1}{1}}\in S^{-1}R.}
9546:if the following conditions are equivalent:
9150:{\displaystyle S=R\setminus {\mathfrak {p}}}
6546:{\displaystyle \operatorname {sat} _{S}(I),}
5252:
5210:
4659:{\displaystyle S=R\setminus {\mathfrak {p}}}
4387:This sort of localization is fundamental in
4249:{\displaystyle S=R\setminus {\mathfrak {p}}}
4045:
4014:
3814:
3808:
3703:
3697:
3089:that is described below. This characterizes
769:{\displaystyle S=R\setminus {\mathfrak {p}}}
677:consists of the powers of a single element,
664:
633:
556:
512:
7503:
3886:is the subset of its elements that are not
2275:{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}
1955:{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}
1427:In the general case, a problem arises with
562:{\displaystyle \{1,s,s^{2},s^{3},\ldots \}}
254:contains information about the behavior of
8997:
8993:
6708:
6704:
6656:
6652:
6628:
6624:
3222:, there exists a unique ring homomorphism
106:is the set of the non-zero elements of an
10090:
9331:
9109:
8259:{\displaystyle S^{-1}M\subseteq S^{-1}N.}
8026:
6888:
6101:
6075:
6004:
5946:. The resulting ring is commonly denoted
5279:if it equals its saturation, that is, if
4883:
4775:Properties to be moved in another section
3760:
3690:
3658:
2630:
1110:
277:
148:
122:
10054:
9927:
9454:§ Localization to Zariski open sets
7399:; if the intersection is nonempty, then
6585:{\displaystyle \operatorname {sat} (I).}
5535:and the universal property implies that
4054:{\displaystyle S=\{1,x,x^{2},\ldots \},}
9815:
9784:is injective (resp. surjective), where
9433:if and only if all its local rings are
5817:; this topological space is called the
2825:{\displaystyle r\mapsto {\frac {r}{1}}}
1422:
670:{\displaystyle S=\{1,t,t^{2},\ldots \}}
14:
10144:
7699:are equivalent if there is an element
3040:The (above defined) ring homomorphism
209:of all functions that are not zero at
7346:{\displaystyle S^{-1}{\mathfrak {p}}}
6126:Localization and saturation of ideals
3377:of, respectively, the multiplicative
3035:
2445:is a commutative ring with addition
2402:{\displaystyle s_{1}r_{2}-s_{2}r_{1}}
2302:is to handle cases such as the above
1404:{\displaystyle {\tfrac {n}{10^{k}}},}
498:{\displaystyle {\tfrac {ab}{s^{2}}}.}
359:
10027:
10000:
9693:The following are local properties:
8949:{\displaystyle \operatorname {Ann} }
8429:, this implies that localization by
8182:
7577:. That is, as a set, it consists of
6176:{\displaystyle j\colon R\to S^{-1}R}
5745:Terminology explained by the context
3997:is an element of a commutative ring
3369:be the categories whose objects are
3260:{\displaystyle g\colon S^{-1}R\to T}
3167:{\displaystyle j\colon R\to S^{-1}R}
3129:The universal property satisfied by
3078:{\displaystyle j\colon R\to S^{-1}R}
2903:does not contain any zero divisors.
9970:Introduction to Commutative Algebra
9879:Localization of a topological space
9855:tag is to do with connections with
9706:is torsion-free (in the case where
9670:
9642:
9604:
9576:
9487:
9475:
9403:
9370:
9349:
9338:
9326:
9299:
9263:
9231:
9166:
9142:
8422:is also an injective homomorphism.
8013:-module with scalar multiplication
7473:
7421:
7381:
7362:
7338:
7286:
7258:
5931:, and is the algebraic analog of a
5911:
5883:
5855:
5072:between the set of prime ideals of
5043:
5031:
5016:
5004:
4960:
4935:
4743:
4679:
4651:
4369:
4334:
4265:
4241:
4194:
3823:{\displaystyle S=R\setminus \{0\},}
879:
853:
792:
761:
391:under multiplication, and contains
173:, as it provides a natural link to
27:Construction of a ring of fractions
24:
9654:{\displaystyle M_{\mathfrak {m}},}
9588:{\displaystyle M_{\mathfrak {p}},}
9415:{\displaystyle R_{\mathfrak {p}}.}
9243:{\displaystyle R_{\mathfrak {p}}.}
9030:
8175:The proof of equivalence (up to a
7300:
6670:
6487:is a multiplicative set in a ring
5631:are two multiplicative sets, then
5605:{\displaystyle {\hat {S}}{}^{-1}R}
5470:So, the images of the elements of
5225:
5103:Saturation of a multiplicative set
4755:{\displaystyle R_{\mathfrak {p}},}
4403:
4346:{\displaystyle R_{\mathfrak {p}},}
3548:
3516:
3484:
3418:
3408:
3398:
3354:
3330:
865:{\displaystyle R_{\mathfrak {p}}.}
348:, and, in particular, that of the
193:defined on some geometric object (
25:
10173:
10125:
9272:{\displaystyle R_{\mathfrak {p}}}
9137:
5920:{\displaystyle R_{\mathfrak {p}}}
4944:{\displaystyle R_{\mathfrak {p}}}
4646:
4236:
3805:
3694:
952:, that consists of the fractions
756:
429:introduces fractions of the form
9310:{\displaystyle {\mathfrak {p}}.}
6553:or, when the multiplicative set
6386:{\displaystyle j^{-1}(S^{-1}I);}
4625:§ Localization of a module
4380:{\displaystyle {\mathfrak {p}}.}
2784:{\displaystyle {\tfrac {1}{1}}.}
2749:{\displaystyle {\tfrac {0}{1}},}
2001:for this relation. The class of
1010:This is a subring since the sum
454:{\displaystyle {\tfrac {a}{s}},}
9839:for a differentiation operator
9678:{\displaystyle {\mathfrak {m}}}
9612:{\displaystyle {\mathfrak {p}}}
9174:{\displaystyle {\mathfrak {p}}}
8998:
8992:
8441:-modules to exact sequences of
8347:
7266:{\displaystyle {\mathfrak {p}}}
7179:
7178:
7033:
6887:
6757:
6748:
6709:
6700:
6657:
6651:
6629:
6623:
5891:{\displaystyle {\mathfrak {p}}}
5863:{\displaystyle {\mathfrak {p}}}
5419:{\displaystyle {\frac {s}{rs}}}
5079:and the set of prime ideals of
4968:{\displaystyle {\mathfrak {p}}}
4687:{\displaystyle {\mathfrak {p}}}
4273:{\displaystyle {\mathfrak {p}}}
4202:{\displaystyle {\mathfrak {p}}}
2036:{\displaystyle {\frac {r}{s}},}
1551:{\displaystyle {\tfrac {a}{1}}}
974:{\displaystyle {\tfrac {a}{s}}}
800:{\displaystyle {\mathfrak {p}}}
418:of all products of elements of
110:, then the localization is the
80:{\displaystyle {\frac {m}{s}},}
10093:Rings and modules of quotients
9945:
9933:
9921:
9912:
9903:
9891:
9777:{\displaystyle f\colon M\to N}
9768:
9024:
9018:
8994:
8917:
8898:
8866:
8863:
8857:
8841:
8787:
8749:
8720:
8717:
8705:
8574:
8571:
8552:
8425:Since the tensor product is a
8377:
8291:{\displaystyle f\colon M\to N}
8282:
7738:
7720:
7601:{\displaystyle {\frac {m}{s}}}
7478:
7468:
7386:
7376:
7234:
7228:
7216:
7210:
7198:
7186:
7134:
7122:
7088:
7082:
7070:
7064:
7052:
7040:
6989:
6977:
6943:
6937:
6925:
6919:
6907:
6895:
6843:
6831:
6791:
6785:
6773:
6767:
6705:
6653:
6648:
6642:
6625:
6576:
6570:
6537:
6531:
6377:
6358:
6154:
5839:The multiplicative set is the
5581:
5483:
5292:
5201:
5149:
4975:can be viewed as a subring of
4538:
4486:
4162:
4147:
4139:
4133:
4124:
4108:
3945:
3665:{\displaystyle R=\mathbb {Z} }
3615:
3612:
3600:
3594:
3591:
3585:
3560:
3557:
3539:
3536:
3533:
3521:
3508:
3496:
3493:
3413:
3362:{\displaystyle {\mathcal {D}}}
3338:{\displaystyle {\mathcal {C}}}
3251:
3203:{\displaystyle f\colon R\to T}
3194:
3145:
3056:
2809:
2263:
2217:
1943:
1897:
1848:
1822:
1816:
1790:
1702:Thus some nonzero elements of
13:
1:
10028:Cohn, P. M. (1991). "§ 9.1".
10001:Cohn, P. M. (1989). "§ 9.3".
9884:
9738:is a commutative domain, and
9118:implies immediately that the
6805:(this is not always true for
6503:; the saturation of an ideal
6082:{\displaystyle p\mathbb {Z} }
5996:, when working over the ring
5133:be a multiplicative set. The
1997:is defined as the set of the
948:of the field of fractions of
205:, then one considers the set
9526:Examples of local properties
6227:the set of the fractions in
6108:{\displaystyle \mathbb {Z} }
6011:{\displaystyle \mathbb {Z} }
5307:{\displaystyle {\hat {S}}=S}
5126:{\displaystyle S\subseteq R}
4560:{\displaystyle R\to S^{-1}R}
4515:is injective if and only if
4508:{\displaystyle R\to S^{-1}R}
3967:{\displaystyle R\to S^{-1}R}
3767:{\displaystyle \mathbb {Q} }
3373:of a commutative ring and a
1753:as above, one considers the
1493:{\displaystyle 0\neq a\in R}
406:. The localization by a set
155:{\displaystyle \mathbb {Z} }
129:{\displaystyle \mathbb {Q} }
7:
9940:Atiyah & Macdonald 1969
9898:Atiyah & Macdonald 1969
9862:
8207:is a multiplicative set in
7750:{\displaystyle u(sn-tm)=0.}
6557:is clear from the context,
4797:{\displaystyle {\sqrt {I}}}
4766:; that is, it has only one
4171:{\displaystyle R=R/(xs-1).}
3638:
3295:{\displaystyle f=g\circ j.}
576:The localization of a ring
366:multiplicatively closed set
291:multiplicatively closed set
270:(c.f. the example given at
10:
10178:
10162:Localization (mathematics)
9874:Localization of a category
7542:localization of the module
5492:{\displaystyle {\hat {S}}}
5158:{\displaystyle {\hat {S}}}
4321:that is generally denoted
3461:{\displaystyle f=g\circ j}
2181:if and only if there is a
1419:is a nonnegative integer.
944:More precisely, it is the
177:theory. In fact, the term
94:belongs to a given subset
8667:finitely presented module
7816:{\displaystyle s,t\in S,}
5799:Hilbert's Nullstellensatz
5785:can be identified with a
4451:{\displaystyle S^{-1}R=0}
4395:. For example, a ring is
1776:{\displaystyle R\times S}
54:, so that it consists of
9900:, Proposition 3.11. (v).
8803:is also an isomorphism.
7848:{\displaystyle m,n\in M}
7504:Localization of a module
6507:by a multiplicative set
6474:{\displaystyle sr\in I.}
6293:{\displaystyle S^{-1}R,}
5528:{\displaystyle S^{-1}R,}
5463:{\displaystyle S^{-1}R.}
5387:{\displaystyle rs\in S,}
4922:. Then its localization
4723:{\displaystyle S^{-1}R,}
3218:(invertible element) in
2888:{\displaystyle S^{-1}R,}
2102:{\displaystyle s^{-1}r.}
1735:{\displaystyle S^{-1}R.}
1317:{\displaystyle S^{-1}R.}
1220:{\displaystyle S^{-1}R.}
613:{\displaystyle S^{-1}R,}
580:by a multiplicative set
379:) of elements of a ring
18:Localization of a module
9962:Atiyah, Michael Francis
9520:Faithfully flat descent
9211:{\displaystyle S^{-1}R}
9100:{\displaystyle t\in S.}
8504:{\displaystyle S^{-1}R}
8467:{\displaystyle S^{-1}R}
7782:{\displaystyle r\in R,}
7668:{\displaystyle s\in S,}
6423:such that there exists
6253:{\displaystyle S^{-1}R}
6220:{\displaystyle S^{-1}I}
5690:{\displaystyle T^{-1}R}
5657:{\displaystyle S^{-1}R}
5561:{\displaystyle S^{-1}R}
5336:{\displaystyle rs\in S}
5314:, or equivalently, if
5271:The multiplicative set
4608:{\displaystyle S^{-1}R}
4310:{\displaystyle S^{-1}R}
4087:{\displaystyle S^{-1}R}
3925:total ring of fractions
3916:{\displaystyle S^{-1}R}
3856:{\displaystyle S^{-1}R}
3745:{\displaystyle S^{-1}R}
3436:Then the factorization
3115:{\displaystyle S^{-1}R}
3026:total ring of fractions
3017:{\displaystyle S^{-1}R}
2961:{\displaystyle S^{-1}R}
2928:{\displaystyle 0\in S,}
2758:multiplicative identity
2438:{\displaystyle S^{-1}R}
1990:{\displaystyle S^{-1}R}
1613:{\displaystyle a\in R,}
1584:{\displaystyle S^{-1}R}
1500:is a zero divisor with
1461:{\displaystyle s\in S,}
1365:{\displaystyle S^{-1}R}
1184:{\displaystyle S^{-1}R}
1003:{\displaystyle s\in S.}
926:{\displaystyle S^{-1}R}
833:{\displaystyle S^{-1}R}
703:{\displaystyle S^{-1}R}
322:{\displaystyle S^{-1}R}
247:{\displaystyle S^{-1}R}
10091:Stenström, Bo (1971).
9918:Matsumura, Theorem 4.7
9845:differential equations
9778:
9742:is a submodule of the
9685:is a maximal ideal of
9679:
9655:
9613:
9589:
9508:faithfully flat module
9497:
9416:
9380:
9311:
9273:
9244:
9212:
9181:in a commutative ring
9175:
9151:
9110:Localization at primes
9101:
9072:
9040:
8950:
8927:
8794:
8659:
8645:is an isomorphism. If
8636:
8505:
8468:
8413:
8292:
8266:This implies that, if
8260:
8166:
8085:
7991:
7927:
7849:
7817:
7783:
7751:
7669:
7640:
7639:{\displaystyle m\in M}
7602:
7494:
7449:
7393:
7347:
7310:
7267:
7241:
7095:
6950:
6798:
6677:
6586:
6547:
6475:
6443:
6442:{\displaystyle s\in S}
6417:
6416:{\displaystyle r\in R}
6387:
6300:which is generated by
6294:
6264:. This is an ideal of
6260:whose numerator is in
6254:
6221:
6177:
6109:
6083:
6012:
5970:
5969:{\displaystyle R_{t},}
5921:
5892:
5864:
5731:belongs to the other.
5721:
5720:{\displaystyle t\in R}
5691:
5658:
5614:canonically isomorphic
5606:
5562:
5529:
5499:are all invertible in
5493:
5464:
5428:multiplicative inverse
5420:
5388:
5362:is not saturated, and
5337:
5308:
5262:
5179:
5159:
5127:
5083:that do not intersect
5053:
4969:
4945:
4891:
4798:
4756:
4724:
4688:
4660:
4609:
4561:
4528:The ring homomorphism
4509:
4452:
4381:
4347:
4311:
4274:
4250:
4213:of a commutative ring
4203:
4172:
4096:canonically isomorphic
4094:can be identified (is
4088:
4055:
3991:
3968:
3917:
3857:
3824:
3768:
3746:
3713:
3666:
3625:
3462:
3427:
3363:
3339:
3321:. More precisely, let
3296:
3261:
3204:
3168:
3116:
3079:
3018:
2962:
2929:
2889:
2853:
2826:
2785:
2750:
2713:
2587:
2439:
2403:
2343:
2296:
2276:
2201:
2200:{\displaystyle t\in S}
2175:
2103:
2067:
2037:
1991:
1956:
1881:
1880:{\displaystyle t\in S}
1855:
1777:
1736:
1696:
1614:
1585:
1552:
1523:
1494:
1462:
1405:
1366:
1318:
1278:
1221:
1185:
1152:
1085:
1004:
975:
927:
866:
834:
801:
770:
734:
733:{\displaystyle R_{t};}
704:
671:
614:
563:
499:
455:
383:, that is a subset of
323:
282:The localization of a
278:Localization of a ring
248:
156:
130:
81:
9822:non-commutative rings
9779:
9680:
9656:
9614:
9590:
9498:
9417:
9381:
9312:
9274:
9245:
9213:
9176:
9152:
9102:
9073:
9041:
8951:
8928:
8795:
8660:
8637:
8506:
8469:
8414:
8293:
8261:
8177:canonical isomorphism
8167:
8086:
7992:
7928:
7850:
7818:
7784:
7752:
7670:
7641:
7603:
7495:
7450:
7394:
7353:is a prime ideal and
7348:
7311:
7268:
7242:
7096:
6951:
6799:
6678:
6587:
6548:
6476:
6444:
6418:
6388:
6295:
6255:
6222:
6178:
6110:
6084:
6013:
5971:
5922:
5893:
5865:
5797:of the ring (this is
5722:
5692:
5659:
5607:
5563:
5530:
5494:
5465:
5421:
5389:
5338:
5309:
5263:
5180:
5160:
5128:
5054:
4970:
4946:
4892:
4799:
4757:
4725:
4689:
4661:
4610:
4562:
4521:does not contain any
4510:
4453:
4382:
4348:
4312:
4275:
4251:
4204:
4173:
4089:
4056:
3992:
3969:
3918:
3858:
3825:
3769:
3747:
3714:
3667:
3626:
3463:
3428:
3364:
3340:
3297:
3262:
3205:
3169:
3117:
3080:
3019:
2963:
2930:
2890:
2854:
2827:
2786:
2751:
2714:
2588:
2440:
2404:
2344:
2297:
2277:
2202:
2176:
2104:
2068:
2038:
1992:
1957:
1882:
1856:
1778:
1737:
1697:
1615:
1586:
1553:
1524:
1522:{\displaystyle as=0.}
1495:
1463:
1406:
1367:
1319:
1279:
1222:
1186:
1153:
1086:
1005:
976:
928:
867:
835:
802:
771:
735:
705:
672:
615:
584:is generally denoted
564:
500:
456:
398:The requirement that
377:multiplicative system
324:
249:
221:. The resulting ring
157:
131:
82:
9816:Non-commutative case
9756:
9665:
9633:
9619:is a prime ideal of
9599:
9567:
9466:
9394:
9321:
9294:
9254:
9222:
9218:is commonly denoted
9189:
9161:
9125:
9114:The definition of a
9082:
9071:{\displaystyle tM=0}
9053:
8967:
8940:
8825:
8676:
8649:
8536:
8482:
8445:
8312:
8270:
8215:
8108:
8020:
7938:
7862:
7827:
7792:
7764:
7714:
7650:
7624:
7585:
7459:
7403:
7357:
7320:
7281:
7253:
7106:
6961:
6815:
6688:
6598:
6561:
6515:
6453:
6427:
6401:
6342:
6268:
6231:
6198:
6142:
6097:
6068:
6000:
5950:
5902:
5878:
5850:
5819:spectrum of the ring
5783:affine algebraic set
5705:
5668:
5635:
5572:
5539:
5503:
5474:
5438:
5398:
5366:
5318:
5283:
5192:
5169:
5140:
5111:
4989:
4955:
4926:
4825:
4784:
4734:
4698:
4674:
4634:
4586:
4532:
4480:
4423:
4364:
4325:
4288:
4260:
4224:
4189:
4102:
4065:
4005:
3981:
3939:
3894:
3834:
3793:
3756:
3723:
3680:
3648:
3475:
3440:
3393:
3349:
3325:
3271:
3226:
3182:
3133:
3093:
3044:
2995:
2939:
2910:
2863:
2843:
2803:
2763:
2728:
2603:
2452:
2416:
2353:
2306:
2286:
2211:
2185:
2113:
2077:
2066:{\displaystyle r/s,}
2046:
2017:
1968:
1891:
1865:
1787:
1761:
1755:equivalence relation
1710:
1624:
1595:
1562:
1533:
1504:
1472:
1443:
1423:General construction
1376:
1343:
1292:
1231:
1195:
1162:
1095:
1014:
985:
956:
933:is a subring of the
904:
844:
811:
787:
744:
714:
681:
624:
588:
509:
465:
433:
333:and denominators in
300:
225:
144:
118:
61:
10060:Commutative algebra
9942:, Proposition 3.14.
9734:(in the case where
9450:algebraic varieties
9435:regular local rings
8427:right exact functor
8303:module homomorphism
7579:equivalence classes
6026:as a property true
5824:In this context, a
4806:radical of an ideal
4389:commutative algebra
3174:is the following:
2976:as unique element.
2282:The reason for the
1999:equivalence classes
1783:that is defined by
1415:is an integer, and
1158:of two elements of
32:commutative algebra
9972:. Westview Press.
9930:, Proposition 2.10
9774:
9744:field of fractions
9712:commutative domain
9675:
9651:
9609:
9585:
9493:
9480:
9412:
9376:
9307:
9285:the local ring of
9269:
9240:
9208:
9171:
9147:
9097:
9068:
9036:
8946:
8923:
8812:finitely generated
8790:
8669:, the natural map
8655:
8632:
8524:universal property
8501:
8464:
8409:
8288:
8256:
8162:
8081:
7987:
7923:
7845:
7813:
7779:
7747:
7665:
7636:
7598:
7522:multiplicative set
7490:
7445:
7389:
7343:
7306:
7263:
7237:
7091:
6946:
6794:
6673:
6582:
6543:
6471:
6439:
6413:
6393:it is an ideal of
6383:
6290:
6250:
6217:
6173:
6105:
6079:
6008:
5994:algebraic topology
5982:neighborhood basis
5966:
5917:
5888:
5860:
5813:equipped with the
5779:algebraic geometry
5717:
5687:
5654:
5602:
5558:
5525:
5489:
5460:
5416:
5384:
5333:
5304:
5258:
5175:
5155:
5123:
5049:
5036:
5009:
4965:
4941:
4887:
4794:
4752:
4720:
4684:
4656:
4605:
4557:
4505:
4448:
4377:
4355:the local ring of
4343:
4307:
4270:
4246:
4199:
4168:
4084:
4051:
3987:
3964:
3913:
3865:field of fractions
3853:
3820:
3764:
3742:
3709:
3662:
3621:
3458:
3423:
3359:
3335:
3292:
3257:
3200:
3164:
3112:
3087:universal property
3075:
3036:Universal property
3014:
2983:is the set of all
2958:
2925:
2885:
2849:
2822:
2781:
2776:
2746:
2741:
2709:
2583:
2435:
2399:
2339:
2334:
2319:
2292:
2272:
2197:
2171:
2169:
2140:
2099:
2063:
2033:
1987:
1952:
1877:
1861:if there exists a
1851:
1773:
1732:
1692:
1687:
1672:
1657:
1637:
1610:
1581:
1548:
1546:
1519:
1490:
1458:
1401:
1396:
1362:
1314:
1274:
1250:
1217:
1181:
1148:
1146:
1121:
1108:
1081:
1076:
1042:
1027:
1000:
971:
969:
935:field of fractions
923:
862:
830:
797:
766:
730:
700:
667:
610:
559:
495:
490:
451:
446:
373:multiplicative set
360:Multiplicative set
346:field of fractions
340:If the ring is an
319:
264:zeros of functions
244:
183:algebraic geometry
171:algebraic geometry
152:
126:
112:field of fractions
77:
36:algebraic geometry
10102:978-3-540-05690-4
10073:978-0-387-94268-1
9979:978-0-201-40751-8
9859:, in particular.
9849:microlocalization
9732:invertible module
9469:
9283:, that is called
9157:of a prime ideal
8658:{\displaystyle M}
8183:Module properties
8076:
8058:
8048:
8035:
7982:
7959:
7949:
7918:
7886:
7873:
7596:
6807:strict inclusions
6760:
6755:
6751:
6712:
6703:
6311:, and called the
5811:topological space
5584:
5486:
5414:
5295:
5204:
5178:{\displaystyle S}
5152:
5025:
4998:
4951:at a prime ideal
4881:
4833:
4792:
4670:of a prime ideal
4573:category of rings
4475:ring homomorphism
3990:{\displaystyle x}
3385:of the ring. The
3319:forgetful functor
2852:{\displaystyle R}
2837:ring homomorphism
2820:
2775:
2740:
2723:additive identity
2704:
2653:
2628:
2578:
2504:
2477:
2412:The localization
2333:
2318:
2295:{\displaystyle t}
2168:
2139:
2028:
1964:The localization
1686:
1671:
1656:
1636:
1545:
1395:
1337:decimal fractions
1335:For example, the
1249:
1145:
1120:
1107:
1075:
1041:
1026:
968:
896:does not contain
710:is often denoted
569:of the powers of
489:
445:
266:that are outside
195:algebraic variety
72:
16:(Redirected from
10169:
10114:
10084:
10051:
10024:
9983:
9952:
9949:
9943:
9937:
9931:
9925:
9919:
9916:
9910:
9907:
9901:
9895:
9791:
9787:
9783:
9781:
9780:
9775:
9749:
9741:
9737:
9729:
9719:
9709:
9705:
9699:
9688:
9684:
9682:
9681:
9676:
9674:
9673:
9660:
9658:
9657:
9652:
9647:
9646:
9645:
9628:
9622:
9618:
9616:
9615:
9610:
9608:
9607:
9594:
9592:
9591:
9586:
9581:
9580:
9579:
9562:
9556:
9552:
9541:
9537:
9533:
9517:
9502:
9500:
9499:
9494:
9492:
9491:
9490:
9479:
9478:
9446:local properties
9442:local properties
9421:
9419:
9418:
9413:
9408:
9407:
9406:
9385:
9383:
9382:
9377:
9375:
9374:
9373:
9363:
9362:
9353:
9352:
9343:
9342:
9341:
9330:
9329:
9317:This means that
9316:
9314:
9313:
9308:
9303:
9302:
9288:
9278:
9276:
9275:
9270:
9268:
9267:
9266:
9249:
9247:
9246:
9241:
9236:
9235:
9234:
9217:
9215:
9214:
9209:
9204:
9203:
9184:
9180:
9178:
9177:
9172:
9170:
9169:
9156:
9154:
9153:
9148:
9146:
9145:
9106:
9104:
9103:
9098:
9077:
9075:
9074:
9069:
9045:
9043:
9042:
9037:
9014:
9013:
8982:
8981:
8955:
8953:
8952:
8947:
8932:
8930:
8929:
8924:
8913:
8912:
8894:
8893:
8889:
8888:
8853:
8852:
8840:
8839:
8799:
8797:
8796:
8791:
8783:
8782:
8764:
8763:
8745:
8744:
8740:
8739:
8701:
8700:
8691:
8690:
8664:
8662:
8661:
8656:
8641:
8639:
8638:
8633:
8628:
8627:
8615:
8614:
8610:
8609:
8589:
8588:
8567:
8566:
8551:
8550:
8516:
8510:
8508:
8507:
8502:
8497:
8496:
8473:
8471:
8470:
8465:
8460:
8459:
8440:
8432:
8418:
8416:
8415:
8410:
8405:
8404:
8392:
8391:
8373:
8372:
8360:
8359:
8340:
8339:
8327:
8326:
8297:
8295:
8294:
8289:
8265:
8263:
8262:
8257:
8249:
8248:
8230:
8229:
8210:
8206:
8202:
8198:
8190:
8171:
8169:
8168:
8163:
8155:
8154:
8142:
8141:
8123:
8122:
8090:
8088:
8087:
8082:
8077:
8072:
8064:
8059:
8051:
8049:
8041:
8036:
8028:
8012:
8008:
7996:
7994:
7993:
7988:
7983:
7981:
7973:
7965:
7960:
7952:
7950:
7942:
7932:
7930:
7929:
7924:
7919:
7917:
7909:
7892:
7887:
7879:
7874:
7866:
7854:
7852:
7851:
7846:
7822:
7820:
7819:
7814:
7788:
7786:
7785:
7780:
7756:
7754:
7753:
7748:
7706:
7702:
7698:
7686:
7674:
7672:
7671:
7666:
7645:
7643:
7642:
7637:
7619:
7607:
7605:
7604:
7599:
7597:
7589:
7576:
7572:
7568:
7559:
7550:
7546:
7535:
7531:
7527:
7519:
7514:commutative ring
7511:
7499:
7497:
7496:
7491:
7477:
7476:
7454:
7452:
7451:
7446:
7441:
7440:
7425:
7424:
7418:
7417:
7398:
7396:
7395:
7390:
7385:
7384:
7366:
7365:
7352:
7350:
7349:
7344:
7342:
7341:
7335:
7334:
7315:
7313:
7312:
7307:
7290:
7289:
7272:
7270:
7269:
7264:
7262:
7261:
7246:
7244:
7243:
7238:
7171:
7170:
7152:
7151:
7121:
7120:
7100:
7098:
7097:
7092:
7026:
7025:
7007:
7006:
6976:
6975:
6955:
6953:
6952:
6947:
6880:
6879:
6861:
6860:
6830:
6829:
6803:
6801:
6800:
6795:
6758:
6756:
6753:
6749:
6744:
6743:
6725:
6724:
6710:
6701:
6682:
6680:
6679:
6674:
6619:
6618:
6591:
6589:
6588:
6583:
6556:
6552:
6550:
6549:
6544:
6527:
6526:
6510:
6506:
6502:
6498:
6494:
6490:
6486:
6480:
6478:
6477:
6472:
6448:
6446:
6445:
6440:
6422:
6420:
6419:
6414:
6396:
6392:
6390:
6389:
6384:
6373:
6372:
6357:
6356:
6337:
6333:
6322:
6318:
6310:
6299:
6297:
6296:
6291:
6283:
6282:
6263:
6259:
6257:
6256:
6251:
6246:
6245:
6226:
6224:
6223:
6218:
6213:
6212:
6193:
6189:
6182:
6180:
6179:
6174:
6169:
6168:
6137:
6133:
6121:
6114:
6112:
6111:
6106:
6104:
6092:
6088:
6086:
6085:
6080:
6078:
6063:
6055:
6051:
6047:
6040:
6032:
6025:
6017:
6015:
6014:
6009:
6007:
5979:
5975:
5973:
5972:
5967:
5962:
5961:
5945:
5941:
5926:
5924:
5923:
5918:
5916:
5915:
5914:
5897:
5895:
5894:
5889:
5887:
5886:
5873:
5869:
5867:
5866:
5861:
5859:
5858:
5815:Zariski topology
5807:commutative ring
5730:
5726:
5724:
5723:
5718:
5700:
5696:
5694:
5693:
5688:
5683:
5682:
5663:
5661:
5660:
5655:
5650:
5649:
5630:
5626:
5619:
5611:
5609:
5608:
5603:
5598:
5597:
5589:
5586:
5585:
5577:
5567:
5565:
5564:
5559:
5554:
5553:
5534:
5532:
5531:
5526:
5518:
5517:
5498:
5496:
5495:
5490:
5488:
5487:
5479:
5469:
5467:
5466:
5461:
5453:
5452:
5433:
5430:of the image of
5425:
5423:
5422:
5417:
5415:
5413:
5402:
5393:
5391:
5390:
5385:
5361:
5354:
5350:
5346:
5342:
5340:
5339:
5334:
5313:
5311:
5310:
5305:
5297:
5296:
5288:
5274:
5267:
5265:
5264:
5259:
5206:
5205:
5197:
5184:
5182:
5181:
5176:
5164:
5162:
5161:
5156:
5154:
5153:
5145:
5132:
5130:
5129:
5124:
5058:
5056:
5055:
5050:
5048:
5047:
5046:
5035:
5034:
5021:
5020:
5019:
5008:
5007:
4974:
4972:
4971:
4966:
4964:
4963:
4950:
4948:
4947:
4942:
4940:
4939:
4938:
4896:
4894:
4893:
4888:
4882:
4877:
4876:
4858:
4850:
4849:
4834:
4829:
4803:
4801:
4800:
4795:
4793:
4788:
4761:
4759:
4758:
4753:
4748:
4747:
4746:
4729:
4727:
4726:
4721:
4713:
4712:
4693:
4691:
4690:
4685:
4683:
4682:
4665:
4663:
4662:
4657:
4655:
4654:
4620:
4614:
4612:
4611:
4606:
4601:
4600:
4566:
4564:
4563:
4558:
4553:
4552:
4520:
4514:
4512:
4511:
4506:
4501:
4500:
4469:
4465:
4457:
4455:
4454:
4449:
4438:
4437:
4386:
4384:
4383:
4378:
4373:
4372:
4358:
4352:
4350:
4349:
4344:
4339:
4338:
4337:
4316:
4314:
4313:
4308:
4303:
4302:
4283:
4279:
4277:
4276:
4271:
4269:
4268:
4255:
4253:
4252:
4247:
4245:
4244:
4216:
4208:
4206:
4205:
4200:
4198:
4197:
4177:
4175:
4174:
4169:
4146:
4123:
4122:
4093:
4091:
4090:
4085:
4080:
4079:
4060:
4058:
4057:
4052:
4038:
4037:
4000:
3996:
3994:
3993:
3988:
3973:
3971:
3970:
3965:
3960:
3959:
3934:
3931:. In this case,
3930:
3922:
3920:
3919:
3914:
3909:
3908:
3885:
3880:commutative ring
3877:
3870:
3862:
3860:
3859:
3854:
3849:
3848:
3829:
3827:
3826:
3821:
3784:
3776:rational numbers
3773:
3771:
3770:
3765:
3763:
3751:
3749:
3748:
3743:
3738:
3737:
3718:
3716:
3715:
3710:
3693:
3671:
3669:
3668:
3663:
3661:
3630:
3628:
3627:
3622:
3575:
3574:
3553:
3552:
3551:
3520:
3519:
3489:
3488:
3487:
3467:
3465:
3464:
3459:
3432:
3430:
3429:
3424:
3422:
3421:
3412:
3411:
3402:
3401:
3368:
3366:
3365:
3360:
3358:
3357:
3344:
3342:
3341:
3336:
3334:
3333:
3301:
3299:
3298:
3293:
3266:
3264:
3263:
3258:
3247:
3246:
3221:
3213:
3209:
3207:
3206:
3201:
3173:
3171:
3170:
3165:
3160:
3159:
3121:
3119:
3118:
3113:
3108:
3107:
3084:
3082:
3081:
3076:
3071:
3070:
3031:
3023:
3021:
3020:
3015:
3010:
3009:
2990:
2985:regular elements
2982:
2975:
2967:
2965:
2964:
2959:
2954:
2953:
2934:
2932:
2931:
2926:
2902:
2894:
2892:
2891:
2886:
2878:
2877:
2858:
2856:
2855:
2850:
2831:
2829:
2828:
2823:
2821:
2813:
2790:
2788:
2787:
2782:
2777:
2768:
2755:
2753:
2752:
2747:
2742:
2733:
2718:
2716:
2715:
2710:
2705:
2703:
2702:
2701:
2692:
2691:
2681:
2680:
2679:
2670:
2669:
2659:
2654:
2652:
2651:
2642:
2641:
2632:
2629:
2627:
2626:
2617:
2616:
2607:
2592:
2590:
2589:
2584:
2579:
2577:
2576:
2575:
2566:
2565:
2555:
2554:
2553:
2544:
2543:
2531:
2530:
2521:
2520:
2510:
2505:
2503:
2502:
2493:
2492:
2483:
2478:
2476:
2475:
2466:
2465:
2456:
2444:
2442:
2441:
2436:
2431:
2430:
2408:
2406:
2405:
2400:
2398:
2397:
2388:
2387:
2375:
2374:
2365:
2364:
2348:
2346:
2345:
2340:
2335:
2326:
2320:
2311:
2301:
2299:
2298:
2293:
2281:
2279:
2278:
2273:
2262:
2261:
2252:
2251:
2239:
2238:
2229:
2228:
2206:
2204:
2203:
2198:
2180:
2178:
2177:
2172:
2170:
2167:
2166:
2157:
2156:
2147:
2141:
2138:
2137:
2128:
2127:
2118:
2108:
2106:
2105:
2100:
2092:
2091:
2072:
2070:
2069:
2064:
2056:
2042:
2040:
2039:
2034:
2029:
2021:
2012:
1996:
1994:
1993:
1988:
1983:
1982:
1961:
1959:
1958:
1953:
1942:
1941:
1932:
1931:
1919:
1918:
1909:
1908:
1886:
1884:
1883:
1878:
1860:
1858:
1857:
1852:
1847:
1846:
1834:
1833:
1815:
1814:
1802:
1801:
1782:
1780:
1779:
1774:
1752:
1748:
1741:
1739:
1738:
1733:
1725:
1724:
1706:must be zero in
1705:
1701:
1699:
1698:
1693:
1688:
1679:
1673:
1664:
1658:
1652:
1644:
1638:
1629:
1619:
1617:
1616:
1611:
1590:
1588:
1587:
1582:
1577:
1576:
1558:is the image in
1557:
1555:
1554:
1549:
1547:
1538:
1528:
1526:
1525:
1520:
1499:
1497:
1496:
1491:
1467:
1465:
1464:
1459:
1438:
1434:
1418:
1414:
1410:
1408:
1407:
1402:
1397:
1394:
1393:
1381:
1371:
1369:
1368:
1363:
1358:
1357:
1327:
1323:
1321:
1320:
1315:
1307:
1306:
1288:is a subring of
1287:
1283:
1281:
1280:
1275:
1267:
1266:
1251:
1242:
1226:
1224:
1223:
1218:
1210:
1209:
1190:
1188:
1187:
1182:
1177:
1176:
1157:
1155:
1154:
1149:
1147:
1144:
1136:
1128:
1122:
1113:
1109:
1100:
1091:and the product
1090:
1088:
1087:
1082:
1077:
1074:
1066:
1049:
1043:
1034:
1028:
1019:
1009:
1007:
1006:
1001:
980:
978:
977:
972:
970:
961:
951:
940:
932:
930:
929:
924:
919:
918:
899:
895:
887:
880:Integral domains
871:
869:
868:
863:
858:
857:
856:
839:
837:
836:
831:
826:
825:
806:
804:
803:
798:
796:
795:
775:
773:
772:
767:
765:
764:
739:
737:
736:
731:
726:
725:
709:
707:
706:
701:
696:
695:
676:
674:
673:
668:
657:
656:
619:
617:
616:
611:
603:
602:
583:
579:
572:
568:
566:
565:
560:
549:
548:
536:
535:
504:
502:
501:
496:
491:
488:
487:
478:
470:
460:
458:
457:
452:
447:
438:
428:
421:
417:
413:
409:
405:
401:
394:
386:
382:
370:
350:rational numbers
336:
332:
328:
326:
325:
320:
315:
314:
295:
288:
284:commutative ring
253:
251:
250:
245:
240:
239:
217:with respect to
161:
159:
158:
153:
151:
138:rational numbers
135:
133:
132:
127:
125:
86:
84:
83:
78:
73:
65:
21:
10177:
10176:
10172:
10171:
10170:
10168:
10167:
10166:
10142:
10141:
10128:
10123:
10103:
10074:
10064:Springer-Verlag
10056:Eisenbud, David
10040:
10013:
9980:
9966:Macdonald, I.G.
9956:
9955:
9950:
9946:
9938:
9934:
9926:
9922:
9917:
9913:
9908:
9904:
9896:
9892:
9887:
9865:
9818:
9810:Noetherian ring
9806:integral domain
9789:
9785:
9757:
9754:
9753:
9747:
9739:
9735:
9727:
9717:
9707:
9703:
9697:
9686:
9669:
9668:
9666:
9663:
9662:
9641:
9640:
9636:
9634:
9631:
9630:
9626:
9620:
9603:
9602:
9600:
9597:
9596:
9575:
9574:
9570:
9568:
9565:
9564:
9560:
9554:
9550:
9539:
9535:
9531:
9528:
9515:
9486:
9485:
9481:
9474:
9473:
9467:
9464:
9463:
9402:
9401:
9397:
9395:
9392:
9391:
9369:
9368:
9364:
9358:
9354:
9348:
9347:
9337:
9336:
9332:
9325:
9324:
9322:
9319:
9318:
9298:
9297:
9295:
9292:
9291:
9286:
9262:
9261:
9257:
9255:
9252:
9251:
9230:
9229:
9225:
9223:
9220:
9219:
9196:
9192:
9190:
9187:
9186:
9182:
9165:
9164:
9162:
9159:
9158:
9141:
9140:
9126:
9123:
9122:
9112:
9083:
9080:
9079:
9054:
9051:
9050:
9009:
9005:
8974:
8970:
8968:
8965:
8964:
8941:
8938:
8937:
8905:
8901:
8881:
8877:
8876:
8872:
8848:
8844:
8832:
8828:
8826:
8823:
8822:
8775:
8771:
8756:
8752:
8732:
8728:
8727:
8723:
8696:
8692:
8683:
8679:
8677:
8674:
8673:
8650:
8647:
8646:
8620:
8616:
8602:
8598:
8597:
8593:
8581:
8577:
8562:
8558:
8543:
8539:
8537:
8534:
8533:
8514:
8489:
8485:
8483:
8480:
8479:
8452:
8448:
8446:
8443:
8442:
8438:
8435:exact sequences
8430:
8400:
8396:
8384:
8380:
8368:
8364:
8352:
8348:
8335:
8331:
8319:
8315:
8313:
8310:
8309:
8271:
8268:
8267:
8241:
8237:
8222:
8218:
8216:
8213:
8212:
8208:
8204:
8200:
8196:
8188:
8185:
8150:
8146:
8134:
8130:
8115:
8111:
8109:
8106:
8105:
8099:tensor products
8065:
8063:
8050:
8040:
8027:
8021:
8018:
8017:
8010:
8001:
7974:
7966:
7964:
7951:
7941:
7939:
7936:
7935:
7910:
7893:
7891:
7878:
7865:
7863:
7860:
7859:
7828:
7825:
7824:
7793:
7790:
7789:
7765:
7762:
7761:
7715:
7712:
7711:
7704:
7700:
7688:
7676:
7651:
7648:
7647:
7625:
7622:
7621:
7609:
7588:
7586:
7583:
7582:
7574:
7570:
7561:
7552:
7548:
7544:
7533:
7529:
7525:
7517:
7509:
7506:
7472:
7471:
7460:
7457:
7456:
7433:
7429:
7420:
7419:
7410:
7406:
7404:
7401:
7400:
7380:
7379:
7361:
7360:
7358:
7355:
7354:
7337:
7336:
7327:
7323:
7321:
7318:
7317:
7285:
7284:
7282:
7279:
7278:
7257:
7256:
7254:
7251:
7250:
7163:
7159:
7144:
7140:
7113:
7109:
7107:
7104:
7103:
7018:
7014:
6999:
6995:
6968:
6964:
6962:
6959:
6958:
6872:
6868:
6853:
6849:
6822:
6818:
6816:
6813:
6812:
6804:
6752:
6736:
6732:
6717:
6713:
6689:
6686:
6685:
6611:
6607:
6599:
6596:
6595:
6562:
6559:
6558:
6554:
6522:
6518:
6516:
6513:
6512:
6508:
6504:
6500:
6496:
6492:
6488:
6484:
6454:
6451:
6450:
6428:
6425:
6424:
6402:
6399:
6398:
6394:
6365:
6361:
6349:
6345:
6343:
6340:
6339:
6335:
6331:
6320:
6316:
6301:
6275:
6271:
6269:
6266:
6265:
6261:
6238:
6234:
6232:
6229:
6228:
6205:
6201:
6199:
6196:
6195:
6191:
6187:
6161:
6157:
6143:
6140:
6139:
6135:
6131:
6128:
6119:
6115:are either the
6100:
6098:
6095:
6094:
6090:
6074:
6069:
6066:
6065:
6061:
6053:
6049:
6045:
6038:
6030:
6023:
6003:
6001:
5998:
5997:
5977:
5957:
5953:
5951:
5948:
5947:
5943:
5939:
5910:
5909:
5905:
5903:
5900:
5899:
5882:
5881:
5879:
5876:
5875:
5871:
5854:
5853:
5851:
5848:
5847:
5791:polynomial ring
5747:
5728:
5706:
5703:
5702:
5698:
5675:
5671:
5669:
5666:
5665:
5642:
5638:
5636:
5633:
5632:
5628:
5624:
5617:
5590:
5588:
5587:
5576:
5575:
5573:
5570:
5569:
5546:
5542:
5540:
5537:
5536:
5510:
5506:
5504:
5501:
5500:
5478:
5477:
5475:
5472:
5471:
5445:
5441:
5439:
5436:
5435:
5431:
5406:
5401:
5399:
5396:
5395:
5367:
5364:
5363:
5359:
5352:
5348:
5344:
5319:
5316:
5315:
5287:
5286:
5284:
5281:
5280:
5272:
5196:
5195:
5193:
5190:
5189:
5170:
5167:
5166:
5144:
5143:
5141:
5138:
5137:
5112:
5109:
5108:
5105:
5042:
5041:
5037:
5030:
5029:
5015:
5014:
5010:
5003:
5002:
4990:
4987:
4986:
4959:
4958:
4956:
4953:
4952:
4934:
4933:
4929:
4927:
4924:
4923:
4901:In particular,
4869:
4865:
4857:
4842:
4838:
4828:
4826:
4823:
4822:
4787:
4785:
4782:
4781:
4742:
4741:
4737:
4735:
4732:
4731:
4705:
4701:
4699:
4696:
4695:
4678:
4677:
4675:
4672:
4671:
4650:
4649:
4635:
4632:
4631:
4618:
4593:
4589:
4587:
4584:
4583:
4545:
4541:
4533:
4530:
4529:
4516:
4493:
4489:
4481:
4478:
4477:
4467:
4461:
4430:
4426:
4424:
4421:
4420:
4406:
4404:Ring properties
4368:
4367:
4365:
4362:
4361:
4356:
4333:
4332:
4328:
4326:
4323:
4322:
4295:
4291:
4289:
4286:
4285:
4281:
4264:
4263:
4261:
4258:
4257:
4240:
4239:
4225:
4222:
4221:
4214:
4193:
4192:
4190:
4187:
4186:
4142:
4115:
4111:
4103:
4100:
4099:
4072:
4068:
4066:
4063:
4062:
4033:
4029:
4006:
4003:
4002:
3998:
3982:
3979:
3978:
3952:
3948:
3940:
3937:
3936:
3932:
3928:
3901:
3897:
3895:
3892:
3891:
3883:
3875:
3868:
3841:
3837:
3835:
3832:
3831:
3794:
3791:
3790:
3787:integral domain
3782:
3759:
3757:
3754:
3753:
3730:
3726:
3724:
3721:
3720:
3689:
3681:
3678:
3677:
3672:is the ring of
3657:
3649:
3646:
3645:
3641:
3567:
3563:
3547:
3546:
3542:
3515:
3514:
3483:
3482:
3478:
3476:
3473:
3472:
3441:
3438:
3437:
3417:
3416:
3407:
3406:
3397:
3396:
3394:
3391:
3390:
3353:
3352:
3350:
3347:
3346:
3329:
3328:
3326:
3323:
3322:
3307:category theory
3272:
3269:
3268:
3239:
3235:
3227:
3224:
3223:
3219:
3211:
3183:
3180:
3179:
3152:
3148:
3134:
3131:
3130:
3100:
3096:
3094:
3091:
3090:
3063:
3059:
3045:
3042:
3041:
3038:
3029:
3002:
2998:
2996:
2993:
2992:
2988:
2980:
2973:
2946:
2942:
2940:
2937:
2936:
2911:
2908:
2907:
2900:
2899:if and only if
2870:
2866:
2864:
2861:
2860:
2844:
2841:
2840:
2812:
2804:
2801:
2800:
2766:
2764:
2761:
2760:
2731:
2729:
2726:
2725:
2697:
2693:
2687:
2683:
2682:
2675:
2671:
2665:
2661:
2660:
2658:
2647:
2643:
2637:
2633:
2631:
2622:
2618:
2612:
2608:
2606:
2604:
2601:
2600:
2596:multiplication
2571:
2567:
2561:
2557:
2556:
2549:
2545:
2539:
2535:
2526:
2522:
2516:
2512:
2511:
2509:
2498:
2494:
2488:
2484:
2482:
2471:
2467:
2461:
2457:
2455:
2453:
2450:
2449:
2423:
2419:
2417:
2414:
2413:
2393:
2389:
2383:
2379:
2370:
2366:
2360:
2356:
2354:
2351:
2350:
2324:
2309:
2307:
2304:
2303:
2287:
2284:
2283:
2257:
2253:
2247:
2243:
2234:
2230:
2224:
2220:
2212:
2209:
2208:
2186:
2183:
2182:
2162:
2158:
2152:
2148:
2145:
2133:
2129:
2123:
2119:
2116:
2114:
2111:
2110:
2084:
2080:
2078:
2075:
2074:
2052:
2047:
2044:
2043:
2020:
2018:
2015:
2014:
2002:
1975:
1971:
1969:
1966:
1965:
1937:
1933:
1927:
1923:
1914:
1910:
1904:
1900:
1892:
1889:
1888:
1866:
1863:
1862:
1842:
1838:
1829:
1825:
1810:
1806:
1797:
1793:
1788:
1785:
1784:
1762:
1759:
1758:
1750:
1746:
1717:
1713:
1711:
1708:
1707:
1703:
1677:
1662:
1645:
1642:
1627:
1625:
1622:
1621:
1596:
1593:
1592:
1569:
1565:
1563:
1560:
1559:
1536:
1534:
1531:
1530:
1505:
1502:
1501:
1473:
1470:
1469:
1444:
1441:
1440:
1439:. Suppose that
1436:
1432:
1425:
1416:
1412:
1389:
1385:
1379:
1377:
1374:
1373:
1350:
1346:
1344:
1341:
1340:
1325:
1299:
1295:
1293:
1290:
1289:
1285:
1259:
1255:
1240:
1232:
1229:
1228:
1202:
1198:
1196:
1193:
1192:
1169:
1165:
1163:
1160:
1159:
1137:
1129:
1126:
1111:
1098:
1096:
1093:
1092:
1067:
1050:
1047:
1032:
1017:
1015:
1012:
1011:
986:
983:
982:
959:
957:
954:
953:
949:
938:
911:
907:
905:
902:
901:
897:
893:
890:integral domain
885:
882:
852:
851:
847:
845:
842:
841:
818:
814:
812:
809:
808:
791:
790:
788:
785:
784:
760:
759:
745:
742:
741:
721:
717:
715:
712:
711:
688:
684:
682:
679:
678:
652:
648:
625:
622:
621:
595:
591:
589:
586:
585:
581:
577:
570:
544:
540:
531:
527:
510:
507:
506:
483:
479:
471:
468:
466:
463:
462:
436:
434:
431:
430:
426:
419:
415:
411:
407:
403:
399:
392:
384:
380:
371:(also called a
368:
362:
342:integral domain
334:
330:
307:
303:
301:
298:
297:
293:
286:
280:
232:
228:
226:
223:
222:
147:
145:
142:
141:
121:
119:
116:
115:
108:integral domain
64:
62:
59:
58:
28:
23:
22:
15:
12:
11:
5:
10175:
10165:
10164:
10159:
10154:
10140:
10139:
10127:
10126:External links
10124:
10122:
10121:
10115:
10101:
10088:
10085:
10072:
10052:
10038:
10025:
10011:
9998:
9984:
9978:
9957:
9954:
9953:
9951:Borel, AG. 3.1
9944:
9932:
9920:
9911:
9909:Borel, AG. 3.3
9902:
9889:
9888:
9886:
9883:
9882:
9881:
9876:
9871:
9869:Local analysis
9864:
9861:
9857:Fourier theory
9817:
9814:
9798:direct product
9794:
9793:
9773:
9770:
9767:
9764:
9761:
9751:
9725:
9715:
9701:
9691:
9690:
9672:
9650:
9644:
9639:
9629:holds for all
9624:
9606:
9584:
9578:
9573:
9563:holds for all
9558:
9544:local property
9527:
9524:
9512:maximal ideals
9504:
9503:
9489:
9484:
9477:
9472:
9427:Nakayama lemma
9411:
9405:
9400:
9386:is the unique
9372:
9367:
9361:
9357:
9351:
9346:
9340:
9335:
9328:
9306:
9301:
9265:
9260:
9239:
9233:
9228:
9207:
9202:
9199:
9195:
9168:
9144:
9139:
9136:
9133:
9130:
9111:
9108:
9096:
9093:
9090:
9087:
9067:
9064:
9061:
9058:
9047:
9046:
9035:
9032:
9029:
9026:
9023:
9020:
9017:
9012:
9008:
9004:
9001:
8996:
8991:
8988:
8985:
8980:
8977:
8973:
8945:
8934:
8933:
8922:
8919:
8916:
8911:
8908:
8904:
8900:
8897:
8892:
8887:
8884:
8880:
8875:
8871:
8868:
8865:
8862:
8859:
8856:
8851:
8847:
8843:
8838:
8835:
8831:
8801:
8800:
8789:
8786:
8781:
8778:
8774:
8770:
8767:
8762:
8759:
8755:
8751:
8748:
8743:
8738:
8735:
8731:
8726:
8722:
8719:
8716:
8713:
8710:
8707:
8704:
8699:
8695:
8689:
8686:
8682:
8654:
8643:
8642:
8631:
8626:
8623:
8619:
8613:
8608:
8605:
8601:
8596:
8592:
8587:
8584:
8580:
8576:
8573:
8570:
8565:
8561:
8557:
8554:
8549:
8546:
8542:
8500:
8495:
8492:
8488:
8463:
8458:
8455:
8451:
8420:
8419:
8408:
8403:
8399:
8395:
8390:
8387:
8383:
8379:
8376:
8371:
8367:
8363:
8358:
8355:
8351:
8346:
8343:
8338:
8334:
8330:
8325:
8322:
8318:
8287:
8284:
8281:
8278:
8275:
8255:
8252:
8247:
8244:
8240:
8236:
8233:
8228:
8225:
8221:
8184:
8181:
8173:
8172:
8161:
8158:
8153:
8149:
8145:
8140:
8137:
8133:
8129:
8126:
8121:
8118:
8114:
8092:
8091:
8080:
8075:
8071:
8068:
8062:
8057:
8054:
8047:
8044:
8039:
8034:
8031:
8025:
7998:
7997:
7986:
7980:
7977:
7972:
7969:
7963:
7958:
7955:
7948:
7945:
7933:
7922:
7916:
7913:
7908:
7905:
7902:
7899:
7896:
7890:
7885:
7882:
7877:
7872:
7869:
7844:
7841:
7838:
7835:
7832:
7812:
7809:
7806:
7803:
7800:
7797:
7778:
7775:
7772:
7769:
7758:
7757:
7746:
7743:
7740:
7737:
7734:
7731:
7728:
7725:
7722:
7719:
7675:and two pairs
7664:
7661:
7658:
7655:
7635:
7632:
7629:
7595:
7592:
7505:
7502:
7501:
7500:
7489:
7486:
7483:
7480:
7475:
7470:
7467:
7464:
7444:
7439:
7436:
7432:
7428:
7423:
7416:
7413:
7409:
7388:
7383:
7378:
7375:
7372:
7369:
7364:
7340:
7333:
7330:
7326:
7305:
7302:
7299:
7296:
7293:
7288:
7260:
7247:
7236:
7233:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7188:
7185:
7182:
7177:
7174:
7169:
7166:
7162:
7158:
7155:
7150:
7147:
7143:
7139:
7136:
7133:
7130:
7127:
7124:
7119:
7116:
7112:
7101:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7063:
7060:
7057:
7054:
7051:
7048:
7045:
7042:
7039:
7036:
7032:
7029:
7024:
7021:
7017:
7013:
7010:
7005:
7002:
6998:
6994:
6991:
6988:
6985:
6982:
6979:
6974:
6971:
6967:
6956:
6945:
6942:
6939:
6936:
6933:
6930:
6927:
6924:
6921:
6918:
6915:
6912:
6909:
6906:
6903:
6900:
6897:
6894:
6891:
6886:
6883:
6878:
6875:
6871:
6867:
6864:
6859:
6856:
6852:
6848:
6845:
6842:
6839:
6836:
6833:
6828:
6825:
6821:
6810:
6793:
6790:
6787:
6784:
6781:
6778:
6775:
6772:
6769:
6766:
6763:
6747:
6742:
6739:
6735:
6731:
6728:
6723:
6720:
6716:
6707:
6699:
6696:
6693:
6683:
6672:
6669:
6666:
6663:
6660:
6655:
6650:
6647:
6644:
6641:
6638:
6635:
6632:
6627:
6622:
6617:
6614:
6610:
6606:
6603:
6581:
6578:
6575:
6572:
6569:
6566:
6542:
6539:
6536:
6533:
6530:
6525:
6521:
6499:are ideals of
6470:
6467:
6464:
6461:
6458:
6438:
6435:
6432:
6412:
6409:
6406:
6382:
6379:
6376:
6371:
6368:
6364:
6360:
6355:
6352:
6348:
6289:
6286:
6281:
6278:
6274:
6249:
6244:
6241:
6237:
6216:
6211:
6208:
6204:
6172:
6167:
6164:
6160:
6156:
6153:
6150:
6147:
6127:
6124:
6103:
6077:
6073:
6006:
5986:
5985:
5965:
5960:
5956:
5936:
5913:
5908:
5885:
5857:
5795:maximal ideals
5746:
5743:
5716:
5713:
5710:
5686:
5681:
5678:
5674:
5653:
5648:
5645:
5641:
5601:
5596:
5593:
5583:
5580:
5557:
5552:
5549:
5545:
5524:
5521:
5516:
5513:
5509:
5485:
5482:
5459:
5456:
5451:
5448:
5444:
5412:
5409:
5405:
5383:
5380:
5377:
5374:
5371:
5332:
5329:
5326:
5323:
5303:
5300:
5294:
5291:
5269:
5268:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5203:
5200:
5174:
5151:
5148:
5122:
5119:
5116:
5104:
5101:
5100:
5099:
5065:
5064:
5061:
5060:
5059:
5045:
5040:
5033:
5028:
5024:
5018:
5013:
5006:
5001:
4997:
4994:
4981:
4980:
4962:
4937:
4932:
4911:
4910:
4899:
4898:
4897:
4886:
4880:
4875:
4872:
4868:
4864:
4861:
4856:
4853:
4848:
4845:
4841:
4837:
4832:
4817:
4816:
4791:
4772:
4771:
4751:
4745:
4740:
4719:
4716:
4711:
4708:
4704:
4681:
4653:
4648:
4645:
4642:
4639:
4628:
4604:
4599:
4596:
4592:
4580:
4575:, that is not
4556:
4551:
4548:
4544:
4540:
4537:
4526:
4504:
4499:
4496:
4492:
4488:
4485:
4471:
4459:if and only if
4447:
4444:
4441:
4436:
4433:
4429:
4405:
4402:
4401:
4400:
4393:local property
4376:
4371:
4342:
4336:
4331:
4306:
4301:
4298:
4294:
4267:
4243:
4238:
4235:
4232:
4229:
4219:set complement
4196:
4183:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4145:
4141:
4138:
4135:
4132:
4129:
4126:
4121:
4118:
4114:
4110:
4107:
4083:
4078:
4075:
4071:
4050:
4047:
4044:
4041:
4036:
4032:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
3986:
3975:
3963:
3958:
3955:
3951:
3947:
3944:
3912:
3907:
3904:
3900:
3872:
3852:
3847:
3844:
3840:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3779:
3762:
3741:
3736:
3733:
3729:
3708:
3705:
3702:
3699:
3696:
3692:
3688:
3685:
3660:
3656:
3653:
3640:
3637:
3632:
3631:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3573:
3570:
3566:
3562:
3559:
3556:
3550:
3545:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3518:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3486:
3481:
3457:
3454:
3451:
3448:
3445:
3420:
3415:
3410:
3405:
3400:
3383:group of units
3356:
3332:
3303:
3302:
3291:
3288:
3285:
3282:
3279:
3276:
3256:
3253:
3250:
3245:
3242:
3238:
3234:
3231:
3199:
3196:
3193:
3190:
3187:
3163:
3158:
3155:
3151:
3147:
3144:
3141:
3138:
3111:
3106:
3103:
3099:
3074:
3069:
3066:
3062:
3058:
3055:
3052:
3049:
3037:
3034:
3024:is called the
3013:
3008:
3005:
3001:
2957:
2952:
2949:
2945:
2924:
2921:
2918:
2915:
2884:
2881:
2876:
2873:
2869:
2848:
2833:
2832:
2819:
2816:
2811:
2808:
2780:
2774:
2771:
2745:
2739:
2736:
2720:
2719:
2708:
2700:
2696:
2690:
2686:
2678:
2674:
2668:
2664:
2657:
2650:
2646:
2640:
2636:
2625:
2621:
2615:
2611:
2594:
2593:
2582:
2574:
2570:
2564:
2560:
2552:
2548:
2542:
2538:
2534:
2529:
2525:
2519:
2515:
2508:
2501:
2497:
2491:
2487:
2481:
2474:
2470:
2464:
2460:
2434:
2429:
2426:
2422:
2396:
2392:
2386:
2382:
2378:
2373:
2369:
2363:
2359:
2338:
2332:
2329:
2323:
2317:
2314:
2291:
2271:
2268:
2265:
2260:
2256:
2250:
2246:
2242:
2237:
2233:
2227:
2223:
2219:
2216:
2196:
2193:
2190:
2165:
2161:
2155:
2151:
2144:
2136:
2132:
2126:
2122:
2098:
2095:
2090:
2087:
2083:
2062:
2059:
2055:
2051:
2032:
2027:
2024:
2013:is denoted as
1986:
1981:
1978:
1974:
1951:
1948:
1945:
1940:
1936:
1930:
1926:
1922:
1917:
1913:
1907:
1903:
1899:
1896:
1876:
1873:
1870:
1850:
1845:
1841:
1837:
1832:
1828:
1824:
1821:
1818:
1813:
1809:
1805:
1800:
1796:
1792:
1772:
1769:
1766:
1731:
1728:
1723:
1720:
1716:
1691:
1685:
1682:
1676:
1670:
1667:
1661:
1655:
1651:
1648:
1641:
1635:
1632:
1609:
1606:
1603:
1600:
1580:
1575:
1572:
1568:
1544:
1541:
1518:
1515:
1512:
1509:
1489:
1486:
1483:
1480:
1477:
1457:
1454:
1451:
1448:
1424:
1421:
1400:
1392:
1388:
1384:
1361:
1356:
1353:
1349:
1313:
1310:
1305:
1302:
1298:
1284:In this case,
1273:
1270:
1265:
1262:
1258:
1254:
1248:
1245:
1239:
1236:
1216:
1213:
1208:
1205:
1201:
1180:
1175:
1172:
1168:
1143:
1140:
1135:
1132:
1125:
1119:
1116:
1106:
1103:
1080:
1073:
1070:
1065:
1062:
1059:
1056:
1053:
1046:
1040:
1037:
1031:
1025:
1022:
999:
996:
993:
990:
967:
964:
922:
917:
914:
910:
884:When the ring
881:
878:
861:
855:
850:
829:
824:
821:
817:
794:
763:
758:
755:
752:
749:
729:
724:
720:
699:
694:
691:
687:
666:
663:
660:
655:
651:
647:
644:
641:
638:
635:
632:
629:
609:
606:
601:
598:
594:
558:
555:
552:
547:
543:
539:
534:
530:
526:
523:
520:
517:
514:
494:
486:
482:
477:
474:
450:
444:
441:
361:
358:
318:
313:
310:
306:
296:is a new ring
279:
276:
243:
238:
235:
231:
213:and localizes
181:originated in
150:
140:from the ring
124:
87:such that the
76:
71:
68:
26:
9:
6:
4:
3:
2:
10174:
10163:
10160:
10158:
10157:Module theory
10155:
10153:
10150:
10149:
10147:
10137:
10133:
10130:
10129:
10119:
10116:
10112:
10108:
10104:
10098:
10094:
10089:
10086:
10083:
10079:
10075:
10069:
10065:
10061:
10057:
10053:
10049:
10045:
10041:
10039:0-471-92840-2
10035:
10031:
10026:
10022:
10018:
10014:
10012:0-471-92234-X
10008:
10004:
9999:
9996:
9995:0-387-97370-2
9992:
9988:
9987:Borel, Armand
9985:
9981:
9975:
9971:
9967:
9963:
9959:
9958:
9948:
9941:
9936:
9929:
9928:Eisenbud 1995
9924:
9915:
9906:
9899:
9894:
9890:
9880:
9877:
9875:
9872:
9870:
9867:
9866:
9860:
9858:
9854:
9850:
9846:
9842:
9838:
9833:
9831:
9830:Ore condition
9827:
9823:
9813:
9811:
9807:
9803:
9799:
9771:
9765:
9762:
9759:
9752:
9745:
9733:
9726:
9723:
9716:
9713:
9702:
9696:
9695:
9694:
9648:
9637:
9625:
9582:
9571:
9559:
9549:
9548:
9547:
9545:
9523:
9521:
9513:
9509:
9482:
9470:
9462:
9461:
9460:
9457:
9455:
9451:
9447:
9443:
9438:
9436:
9432:
9428:
9422:
9409:
9398:
9389:
9388:maximal ideal
9365:
9359:
9355:
9344:
9333:
9304:
9289:
9282:
9258:
9237:
9226:
9205:
9200:
9197:
9193:
9134:
9131:
9128:
9121:
9117:
9107:
9094:
9091:
9088:
9085:
9065:
9062:
9059:
9056:
9033:
9027:
9021:
9015:
9010:
9006:
9002:
8999:
8989:
8986:
8983:
8978:
8975:
8971:
8963:
8962:
8961:
8959:
8943:
8920:
8914:
8909:
8906:
8902:
8895:
8890:
8885:
8882:
8878:
8873:
8869:
8860:
8854:
8849:
8845:
8836:
8833:
8829:
8821:
8820:
8819:
8817:
8813:
8809:
8804:
8784:
8779:
8776:
8772:
8768:
8765:
8760:
8757:
8753:
8746:
8741:
8736:
8733:
8729:
8724:
8714:
8711:
8708:
8702:
8697:
8693:
8687:
8684:
8680:
8672:
8671:
8670:
8668:
8652:
8629:
8624:
8621:
8617:
8611:
8606:
8603:
8599:
8594:
8590:
8585:
8582:
8578:
8568:
8563:
8559:
8555:
8547:
8544:
8540:
8532:
8531:
8530:
8529:
8525:
8520:
8518:
8498:
8493:
8490:
8486:
8477:
8476:exact functor
8461:
8456:
8453:
8449:
8436:
8428:
8423:
8406:
8401:
8397:
8393:
8388:
8385:
8381:
8374:
8369:
8365:
8361:
8356:
8353:
8349:
8344:
8341:
8336:
8332:
8328:
8323:
8320:
8316:
8308:
8307:
8306:
8304:
8301:
8285:
8279:
8276:
8273:
8253:
8250:
8245:
8242:
8238:
8234:
8231:
8226:
8223:
8219:
8194:
8180:
8178:
8159:
8156:
8151:
8147:
8143:
8138:
8135:
8131:
8127:
8124:
8119:
8116:
8112:
8104:
8103:
8102:
8100:
8095:
8078:
8073:
8069:
8066:
8060:
8055:
8052:
8045:
8042:
8037:
8032:
8029:
8023:
8016:
8015:
8014:
8007:
8004:
7984:
7978:
7975:
7970:
7967:
7961:
7956:
7953:
7946:
7943:
7934:
7920:
7914:
7911:
7906:
7903:
7900:
7897:
7894:
7888:
7883:
7880:
7875:
7870:
7867:
7858:
7857:
7856:
7842:
7839:
7836:
7833:
7830:
7810:
7807:
7804:
7801:
7798:
7795:
7776:
7773:
7770:
7767:
7744:
7741:
7735:
7732:
7729:
7726:
7723:
7717:
7710:
7709:
7708:
7696:
7692:
7684:
7680:
7662:
7659:
7656:
7653:
7633:
7630:
7627:
7617:
7613:
7593:
7590:
7580:
7567:
7564:
7558:
7555:
7543:
7539:
7523:
7515:
7487:
7484:
7481:
7465:
7462:
7442:
7437:
7434:
7430:
7426:
7414:
7411:
7407:
7373:
7370:
7367:
7331:
7328:
7324:
7303:
7297:
7294:
7291:
7276:
7248:
7231:
7225:
7222:
7219:
7213:
7207:
7204:
7201:
7195:
7192:
7189:
7183:
7180:
7175:
7172:
7167:
7164:
7160:
7156:
7153:
7148:
7145:
7141:
7137:
7131:
7128:
7125:
7117:
7114:
7110:
7102:
7085:
7079:
7076:
7073:
7067:
7061:
7058:
7055:
7049:
7046:
7043:
7037:
7034:
7030:
7027:
7022:
7019:
7015:
7011:
7008:
7003:
7000:
6996:
6992:
6986:
6983:
6980:
6972:
6969:
6965:
6957:
6940:
6934:
6931:
6928:
6922:
6916:
6913:
6910:
6904:
6901:
6898:
6892:
6889:
6884:
6881:
6876:
6873:
6869:
6865:
6862:
6857:
6854:
6850:
6846:
6840:
6837:
6834:
6826:
6823:
6819:
6811:
6808:
6788:
6782:
6779:
6776:
6770:
6764:
6761:
6745:
6740:
6737:
6733:
6729:
6726:
6721:
6718:
6714:
6697:
6694:
6691:
6684:
6667:
6664:
6661:
6658:
6645:
6639:
6636:
6633:
6630:
6620:
6615:
6612:
6608:
6604:
6601:
6594:
6593:
6592:
6579:
6573:
6567:
6564:
6540:
6534:
6528:
6523:
6519:
6481:
6468:
6465:
6462:
6459:
6456:
6436:
6433:
6430:
6410:
6407:
6404:
6380:
6374:
6369:
6366:
6362:
6353:
6350:
6346:
6329:
6324:
6314:
6308:
6304:
6287:
6284:
6279:
6276:
6272:
6247:
6242:
6239:
6235:
6214:
6209:
6206:
6202:
6186:
6170:
6165:
6162:
6158:
6151:
6148:
6145:
6123:
6118:
6117:singleton set
6071:
6059:
6044:
6036:
6029:
6021:
5995:
5991:
5990:number theory
5983:
5963:
5958:
5954:
5937:
5934:
5933:ring of germs
5930:
5906:
5846:
5842:
5838:
5837:
5836:
5833:
5831:
5827:
5822:
5820:
5816:
5812:
5808:
5804:
5800:
5796:
5792:
5788:
5787:quotient ring
5784:
5780:
5776:
5772:
5768:
5764:
5760:
5756:
5752:
5742:
5741:of the ring.
5740:
5737:
5732:
5714:
5711:
5708:
5684:
5679:
5676:
5672:
5651:
5646:
5643:
5639:
5621:
5615:
5599:
5594:
5591:
5578:
5555:
5550:
5547:
5543:
5522:
5519:
5514:
5511:
5507:
5480:
5457:
5454:
5449:
5446:
5442:
5429:
5410:
5407:
5403:
5381:
5378:
5375:
5372:
5369:
5356:
5343:implies that
5330:
5327:
5324:
5321:
5301:
5298:
5289:
5278:
5255:
5249:
5246:
5243:
5240:
5237:
5234:
5231:
5228:
5222:
5219:
5216:
5213:
5207:
5198:
5188:
5187:
5186:
5172:
5146:
5136:
5120:
5117:
5114:
5097:
5094:
5090:
5086:
5082:
5078:
5075:
5071:
5067:
5066:
5062:
5038:
5026:
5022:
5011:
4999:
4995:
4992:
4985:
4984:
4983:
4982:
4978:
4930:
4921:
4917:
4913:
4912:
4908:
4904:
4900:
4884:
4878:
4873:
4870:
4866:
4862:
4859:
4854:
4851:
4846:
4843:
4839:
4835:
4830:
4821:
4820:
4819:
4818:
4814:
4810:
4807:
4789:
4779:
4778:
4777:
4776:
4769:
4768:maximal ideal
4765:
4749:
4738:
4717:
4714:
4709:
4706:
4702:
4669:
4643:
4640:
4637:
4629:
4627:for details).
4626:
4622:
4602:
4597:
4594:
4590:
4581:
4578:
4574:
4570:
4554:
4549:
4546:
4542:
4535:
4527:
4524:
4523:zero divisors
4519:
4502:
4497:
4494:
4490:
4483:
4476:
4472:
4464:
4460:
4445:
4442:
4439:
4434:
4431:
4427:
4419:
4418:
4417:
4415:
4411:
4398:
4394:
4390:
4374:
4360:
4340:
4329:
4320:
4304:
4299:
4296:
4292:
4233:
4230:
4227:
4220:
4212:
4184:
4181:
4180:affine scheme
4165:
4159:
4156:
4153:
4150:
4143:
4136:
4130:
4127:
4119:
4116:
4112:
4105:
4097:
4081:
4076:
4073:
4069:
4048:
4042:
4039:
4034:
4030:
4026:
4023:
4020:
4017:
4011:
4008:
3984:
3976:
3961:
3956:
3953:
3949:
3942:
3926:
3910:
3905:
3902:
3898:
3889:
3888:zero divisors
3881:
3873:
3866:
3850:
3845:
3842:
3838:
3817:
3811:
3802:
3799:
3796:
3788:
3780:
3777:
3752:is the field
3739:
3734:
3731:
3727:
3706:
3700:
3686:
3683:
3675:
3654:
3651:
3643:
3642:
3636:
3618:
3609:
3606:
3603:
3597:
3588:
3582:
3579:
3576:
3571:
3568:
3564:
3554:
3543:
3530:
3527:
3524:
3511:
3505:
3502:
3499:
3490:
3479:
3471:
3470:
3469:
3455:
3452:
3449:
3446:
3443:
3434:
3403:
3388:
3384:
3380:
3376:
3372:
3320:
3316:
3312:
3308:
3289:
3286:
3283:
3280:
3277:
3274:
3254:
3248:
3243:
3240:
3236:
3232:
3229:
3217:
3197:
3191:
3188:
3185:
3177:
3176:
3175:
3161:
3156:
3153:
3149:
3142:
3139:
3136:
3127:
3125:
3109:
3104:
3101:
3097:
3088:
3072:
3067:
3064:
3060:
3053:
3050:
3047:
3033:
3027:
3011:
3006:
3003:
2999:
2986:
2977:
2971:
2955:
2950:
2947:
2943:
2922:
2919:
2916:
2913:
2904:
2898:
2882:
2879:
2874:
2871:
2867:
2846:
2838:
2817:
2814:
2806:
2799:
2798:
2797:
2796:
2791:
2778:
2772:
2769:
2759:
2743:
2737:
2734:
2724:
2706:
2698:
2694:
2688:
2684:
2676:
2672:
2666:
2662:
2655:
2648:
2644:
2638:
2634:
2623:
2619:
2613:
2609:
2599:
2598:
2597:
2580:
2572:
2568:
2562:
2558:
2550:
2546:
2540:
2536:
2532:
2527:
2523:
2517:
2513:
2506:
2499:
2495:
2489:
2485:
2479:
2472:
2468:
2462:
2458:
2448:
2447:
2446:
2432:
2427:
2424:
2420:
2410:
2394:
2390:
2384:
2380:
2376:
2371:
2367:
2361:
2357:
2336:
2330:
2327:
2321:
2315:
2312:
2289:
2269:
2266:
2258:
2254:
2248:
2244:
2240:
2235:
2231:
2225:
2221:
2214:
2194:
2191:
2188:
2163:
2159:
2153:
2149:
2142:
2134:
2130:
2124:
2120:
2096:
2093:
2088:
2085:
2081:
2060:
2057:
2053:
2049:
2030:
2025:
2022:
2010:
2006:
2000:
1984:
1979:
1976:
1972:
1962:
1949:
1946:
1938:
1934:
1928:
1924:
1920:
1915:
1911:
1905:
1901:
1894:
1874:
1871:
1868:
1843:
1839:
1835:
1830:
1826:
1819:
1811:
1807:
1803:
1798:
1794:
1770:
1767:
1764:
1756:
1743:
1729:
1726:
1721:
1718:
1714:
1689:
1683:
1680:
1674:
1668:
1665:
1659:
1653:
1649:
1646:
1639:
1633:
1630:
1607:
1604:
1601:
1598:
1578:
1573:
1570:
1566:
1542:
1539:
1516:
1513:
1510:
1507:
1487:
1484:
1481:
1478:
1475:
1455:
1452:
1449:
1446:
1430:
1429:zero divisors
1420:
1398:
1390:
1386:
1382:
1359:
1354:
1351:
1347:
1338:
1333:
1331:
1330:zero divisors
1311:
1308:
1303:
1300:
1296:
1271:
1268:
1263:
1260:
1256:
1252:
1246:
1243:
1237:
1234:
1214:
1211:
1206:
1203:
1199:
1178:
1173:
1170:
1166:
1141:
1138:
1133:
1130:
1123:
1117:
1114:
1104:
1101:
1078:
1071:
1068:
1063:
1060:
1057:
1054:
1051:
1044:
1038:
1035:
1029:
1023:
1020:
997:
994:
991:
988:
965:
962:
947:
942:
936:
920:
915:
912:
908:
891:
877:
876:
872:
859:
848:
827:
822:
819:
815:
783:
779:
753:
750:
747:
727:
722:
718:
697:
692:
689:
685:
661:
658:
653:
649:
645:
642:
639:
636:
630:
627:
607:
604:
599:
596:
592:
574:
553:
550:
545:
541:
537:
532:
528:
524:
521:
518:
515:
492:
484:
480:
475:
472:
448:
442:
439:
423:
396:
390:
378:
374:
367:
357:
355:
354:zero divisors
351:
347:
343:
338:
316:
311:
308:
304:
292:
285:
275:
273:
269:
265:
261:
257:
241:
236:
233:
229:
220:
216:
212:
208:
204:
200:
196:
192:
189:is a ring of
188:
184:
180:
176:
172:
167:
165:
139:
113:
109:
105:
101:
97:
93:
90:
74:
69:
66:
57:
53:
49:
45:
41:
37:
33:
19:
10132:Localization
10092:
10059:
10029:
10002:
9969:
9947:
9935:
9923:
9914:
9905:
9893:
9852:
9840:
9836:
9834:
9825:
9819:
9795:
9692:
9543:
9529:
9518:). See also
9505:
9458:
9441:
9439:
9423:
9390:of the ring
9284:
9113:
9049:that is, if
9048:
8935:
8815:
8807:
8806:If a module
8805:
8802:
8644:
8521:
8424:
8421:
8186:
8174:
8096:
8093:
8005:
8002:
7999:
7759:
7694:
7690:
7682:
7678:
7615:
7611:
7565:
7562:
7556:
7553:
7541:
7507:
6482:
6327:
6325:
6313:localization
6312:
6306:
6302:
6129:
6058:prime number
6042:
6034:
6027:
5987:
5834:
5829:
5826:localization
5825:
5823:
5803:prime ideals
5762:
5751:localization
5750:
5748:
5735:
5733:
5622:
5357:
5276:
5270:
5185:is the set
5134:
5106:
5095:
5092:
5088:
5084:
5080:
5076:
5073:
4976:
4919:
4915:
4902:
4812:
4808:
4774:
4773:
4517:
4462:
4407:
4354:
3633:
3435:
3315:left adjoint
3304:
3128:
3085:satisfies a
3039:
2978:
2905:
2834:
2792:
2721:
2595:
2411:
2109:So, one has
2008:
2004:
1963:
1744:
1620:and one has
1426:
1334:
943:
883:
874:
873:
575:
424:
397:
376:
372:
363:
339:
281:
267:
259:
255:
218:
214:
210:
206:
202:
198:
186:
179:localization
178:
168:
103:
99:
95:
91:
51:
40:localization
39:
29:
10152:Ring theory
9820:Localizing
9788:is another
9722:flat module
9530:A property
9116:prime ideal
8958:annihilator
8528:natural map
8009:is also an
7608:, of pairs
7275:prime ideal
6511:is denoted
5845:prime ideal
5759:topological
5755:geometrical
5068:There is a
4979:. Moreover,
4804:denote the
4579:in general.
4569:epimorphism
4353:and called
4211:prime ideal
3124:isomorphism
900:, the ring
840:is denoted
782:prime ideal
89:denominator
10146:Categories
10118:Serge Lang
9885:References
9804:is not an
9553:holds for
9456:, below.)
9281:local ring
9120:complement
8818:, one has
8211:, one has
8000:Moreover,
7707:such that
7581:, denoted
7551:, denoted
7277:such that
6328:saturation
6052:, and, if
5942:of a ring
5929:local ring
5870:of a ring
5841:complement
5727:such that
5135:saturation
4764:local ring
4668:complement
4577:surjective
4319:local ring
3267:such that
2835:defines a
2207:such that
1887:such that
981:such that
778:complement
272:local ring
10136:MathWorld
9769:→
9763::
9471:⨁
9356:⊗
9250:The ring
9198:−
9138:∖
9089:∈
9078:for some
9031:∅
9028:≠
9016:
9003:∩
8995:⟺
8976:−
8907:−
8896:
8883:−
8855:
8834:−
8777:−
8758:−
8747:
8734:−
8721:→
8703:
8685:−
8622:−
8604:−
8595:⊗
8583:−
8575:→
8560:⊗
8545:−
8491:−
8454:−
8398:⊗
8386:−
8378:→
8366:⊗
8354:−
8333:⊗
8321:−
8300:injective
8283:→
8277::
8243:−
8235:⊆
8224:−
8193:submodule
8148:⊗
8136:−
8117:−
7840:∈
7805:∈
7771:∈
7730:−
7657:∈
7631:∈
7466:
7435:−
7412:−
7374:
7329:−
7301:∅
7292:∩
7226:
7220:⋅
7208:
7193:⋅
7184:
7165:−
7157:⋅
7146:−
7129:⋅
7115:−
7080:
7062:
7038:
7020:−
7001:−
6970:−
6935:
6929:∩
6917:
6902:∩
6893:
6874:−
6866:∩
6855:−
6838:∩
6824:−
6783:
6777:⊆
6765:
6738:−
6730:⊆
6719:−
6706:⟹
6695:⊆
6671:∅
6668:≠
6662:∩
6654:⟺
6640:
6634:∈
6626:⟺
6613:−
6605:∈
6568:
6529:
6463:∈
6434:∈
6408:∈
6367:−
6351:−
6277:−
6240:−
6207:−
6163:−
6155:→
6149::
6043:Away from
5767:manifolds
5749:The term
5712:∈
5677:−
5644:−
5592:−
5582:^
5548:−
5512:−
5484:^
5447:−
5376:∈
5328:∈
5293:^
5277:saturated
5247:∈
5232:∈
5226:∃
5223::
5217:∈
5202:^
5150:^
5118:⊆
5070:bijection
5027:⋂
5000:⋂
4871:−
4863:⋅
4844:−
4836:⋅
4707:−
4647:∖
4595:−
4582:The ring
4547:−
4539:→
4495:−
4487:→
4466:contains
4432:−
4297:−
4237:∖
4157:−
4117:−
4074:−
4043:…
3954:−
3946:→
3903:−
3882:, and if
3843:−
3806:∖
3732:−
3695:∖
3569:−
3555:
3540:→
3491:
3453:∘
3414:→
3404::
3387:morphisms
3375:submonoid
3284:∘
3252:→
3241:−
3233::
3195:→
3189::
3154:−
3146:→
3140::
3122:up to an
3102:−
3065:−
3057:→
3051::
3004:−
2972:that has
2970:zero ring
2948:−
2917:∈
2897:injective
2895:which is
2872:−
2810:↦
2425:−
2377:−
2241:−
2192:∈
2086:−
1977:−
1921:−
1872:∈
1820:∼
1768:×
1719:−
1602:∈
1571:−
1485:∈
1479:≠
1450:∈
1352:−
1328:contains
1301:−
1261:−
1253:∈
1204:−
1171:−
992:∈
913:−
820:−
757:∖
690:−
662:…
597:−
554:…
309:−
234:−
191:functions
56:fractions
10058:(1995),
9968:(1969).
9863:See also
9792:-module.
9700:is zero.
9538:-module
8956:denotes
8305:, then
8199:-module
7620:, where
7560:, is an
6020:integers
5761:objects
4730:denoted
3674:integers
3639:Examples
3313:that is
2795:function
387:that is
164:integers
10111:0325663
10082:1322960
10048:1098018
10030:Algebra
10021:1006872
10003:Algebra
9431:regular
8517:-module
5763:locally
5351:are in
4907:reduced
4694:, then
4666:is the
4621:-module
4571:in the
4414:modules
4397:regular
3923:is the
3890:, then
3863:is the
3774:of the
3381:or the
3311:functor
2968:is the
1191:are in
946:subring
807:, then
776:is the
10109:
10099:
10080:
10070:
10046:
10036:
10019:
10009:
9993:
9976:
9853:micro-
9808:nor a
9802:fields
9730:is an
9720:is a
9661:where
9595:where
9534:of an
8936:where
8478:, and
8298:is an
8203:, and
8195:of an
7540:. The
7538:module
7532:be an
7528:, and
6759:
6750:
6711:
6702:
6491:, and
6194:, let
6138:, and
6060:, "at
5830:points
5775:sheafs
4815:, then
4567:is an
4410:ideals
4217:, the
3789:, and
3785:is an
3676:, and
3379:monoid
3305:Using
2349:where
1745:Given
1431:. Let
1411:where
888:is an
389:closed
48:module
10134:from
9710:is a
9542:is a
9506:is a
9279:is a
8814:over
8810:is a
8665:is a
8513:flat
8511:is a
8433:maps
8191:is a
7520:be a
7512:be a
7316:then
7273:is a
6449:with
6185:ideal
6056:is a
6037:from
5927:is a
5843:of a
5805:of a
5789:of a
5781:, an
5777:. In
5771:germs
5739:units
5426:is a
5394:then
4762:is a
4623:(see
4617:flat
4615:is a
4317:is a
4209:is a
4061:then
3878:is a
3830:then
3719:then
3371:pairs
3317:to a
3214:to a
2935:then
2859:into
2839:from
1529:Then
780:of a
375:or a
289:by a
258:near
185:: if
175:sheaf
102:. If
10097:ISBN
10068:ISBN
10034:ISBN
10007:ISBN
9991:ISBN
9974:ISBN
7823:and
7687:and
7646:and
7508:Let
7455:and
6495:and
6326:The
6130:Let
6035:away
5992:and
5773:and
5757:and
5664:and
5627:and
5612:are
5568:and
5347:and
5107:Let
4914:Let
4473:The
4098:to)
4001:and
3345:and
3216:unit
3178:If
2793:The
2756:and
1749:and
1468:and
892:and
44:ring
34:and
9800:of
9746:of
9514:of
9448:of
9290:at
9007:Ann
8944:Ann
8874:Ann
8846:Ann
8725:Hom
8694:Hom
8437:of
8187:If
7855:):
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7547:by
7524:in
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7371:sat
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7077:sat
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7035:sat
6932:sat
6914:sat
6890:sat
6780:sat
6762:sat
6754:and
6637:sat
6565:sat
6520:sat
6338:is
6334:by
6330:of
6319:by
6315:of
6190:in
6120:{p}
6033:or
6018:of
5988:In
5736:all
5623:If
5434:in
5358:If
5275:is
5165:of
4905:is
4811:in
4630:If
4280:in
4256:of
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3977:If
3927:of
3874:If
3867:of
3781:If
3644:If
3544:hom
3480:hom
3028:of
2987:of
2979:If
2906:If
2073:or
1757:on
1591:of
937:of
740:if
274:).
162:of
136:of
98:of
46:or
30:In
10148::
10107:MR
10105:.
10078:MR
10076:,
10066:,
10044:MR
10042:.
10017:MR
10015:.
9964:;
9832:.
9750:).
9714:).
9522:.
9437:.
8519:.
8101::
7745:0.
7693:,
7681:,
7614:,
7516:,
6323:.
6028:at
5821:.
5809:a
5769:,
5729:st
5620:.
5355:.
5091:→
4412:,
4359:at
3032:.
2270:0.
2007:,
1950:0.
1517:0.
1387:10
1332:.
395:.
337:.
197:)
166:.
38:,
10138:.
10113:.
10050:.
10023:.
9997:.
9982:.
9841:D
9837:D
9826:S
9790:R
9786:N
9772:N
9766:M
9760:f
9748:R
9740:M
9736:R
9728:M
9724:.
9718:M
9708:R
9704:M
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9689:.
9687:R
9671:m
9649:,
9643:m
9638:M
9627:P
9623:.
9621:R
9605:p
9583:,
9577:p
9572:M
9561:P
9557:.
9555:M
9551:P
9540:M
9536:R
9532:P
9516:R
9488:p
9483:R
9476:p
9410:.
9404:p
9399:R
9371:p
9366:R
9360:R
9350:p
9345:=
9339:p
9334:R
9327:p
9305:.
9300:p
9287:R
9264:p
9259:R
9238:.
9232:p
9227:R
9206:R
9201:1
9194:S
9183:R
9167:p
9143:p
9135:R
9132:=
9129:S
9095:.
9092:S
9086:t
9066:0
9063:=
9060:M
9057:t
9034:,
9025:)
9022:M
9019:(
9011:R
9000:S
8990:0
8987:=
8984:M
8979:1
8972:S
8921:,
8918:)
8915:M
8910:1
8903:S
8899:(
8891:R
8886:1
8879:S
8870:=
8867:)
8864:)
8861:M
8858:(
8850:R
8842:(
8837:1
8830:S
8816:R
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8788:)
8785:N
8780:1
8773:S
8769:,
8766:M
8761:1
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8750:(
8742:R
8737:1
8730:S
8718:)
8715:N
8712:,
8709:M
8706:(
8698:R
8688:1
8681:S
8653:M
8630:N
8625:1
8618:S
8612:R
8607:1
8600:S
8591:M
8586:1
8579:S
8572:)
8569:N
8564:R
8556:M
8553:(
8548:1
8541:S
8515:R
8499:R
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8462:R
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8450:S
8439:R
8431:S
8407:N
8402:R
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8375:M
8370:R
8362:R
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8350:S
8345::
8342:f
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8317:S
8286:N
8280:M
8274:f
8254:.
8251:N
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8239:S
8232:M
8227:1
8220:S
8209:R
8205:S
8201:N
8197:R
8189:M
8160:.
8157:M
8152:R
8144:R
8139:1
8132:S
8128:=
8125:M
8120:1
8113:S
8079:.
8074:s
8070:m
8067:r
8061:=
8056:s
8053:m
8046:1
8043:r
8038:=
8033:s
8030:m
8024:r
8011:R
8006:M
8003:S
7985:.
7979:t
7976:s
7971:m
7968:r
7962:=
7957:t
7954:m
7947:s
7944:r
7921:,
7915:t
7912:s
7907:n
7904:s
7901:+
7898:m
7895:t
7889:=
7884:t
7881:n
7876:+
7871:s
7868:m
7843:M
7837:n
7834:,
7831:m
7811:,
7808:S
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7799:,
7796:s
7777:,
7774:R
7768:r
7742:=
7739:)
7736:m
7733:t
7727:n
7724:s
7721:(
7718:u
7705:S
7701:u
7697:)
7695:t
7691:n
7689:(
7685:)
7683:s
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7677:(
7663:,
7660:S
7654:s
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7628:m
7618:)
7616:s
7612:m
7610:(
7594:s
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7575:M
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7566:R
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7554:S
7549:S
7545:M
7536:-
7534:R
7530:M
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7510:R
7488:.
7485:R
7482:=
7479:)
7474:p
7469:(
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7427:=
7422:p
7415:1
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7387:)
7382:p
7377:(
7368:=
7363:p
7339:p
7332:1
7325:S
7304:,
7298:=
7295:S
7287:p
7259:p
7235:)
7232:J
7229:(
7217:)
7214:I
7211:(
7202:=
7199:)
7196:J
7190:I
7187:(
7176:,
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7149:1
7142:S
7138:=
7135:)
7132:J
7126:I
7123:(
7118:1
7111:S
7089:)
7086:J
7083:(
7074:+
7071:)
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7065:(
7056:=
7053:)
7050:J
7047:+
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7041:(
7031:,
7028:J
7023:1
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7004:1
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6993:=
6990:)
6987:J
6984:+
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6973:1
6966:S
6944:)
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6926:)
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6911:=
6908:)
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6847:=
6844:)
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6792:)
6789:J
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6774:)
6771:I
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6631:1
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6616:1
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6580:.
6577:)
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6046:n
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6031:n
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6005:Z
5978:t
5964:,
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5944:R
5940:t
5935:.
5912:p
5907:R
5884:p
5872:R
5856:p
5715:R
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5699:s
5685:R
5680:1
5673:T
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5640:S
5629:T
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5600:R
5595:1
5579:S
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5551:1
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5520:R
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5458:.
5455:R
5450:1
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5411:s
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5360:S
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5349:s
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5299:=
5290:S
5273:S
5256:.
5253:}
5250:S
5244:s
5241:r
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5211:{
5208:=
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5023:=
5017:p
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513:{
493:.
485:2
481:s
476:b
473:a
449:,
443:s
440:a
427:s
420:U
416:S
412:U
408:U
404:S
400:S
393:1
385:R
381:R
369:S
335:S
331:R
317:R
312:1
305:S
294:S
287:R
268:V
260:p
256:V
242:R
237:1
230:S
219:S
215:R
211:p
207:S
203:p
199:V
187:R
149:Z
123:Q
104:S
100:R
96:S
92:s
75:,
70:s
67:m
52:R
20:)
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