1788:
Because noncommutative rings of scientific interest are more complicated than commutative rings, their structure, properties and behavior are less well understood. A great deal of work has been done successfully generalizing some results from commutative rings to noncommutative rings. A major
1739:
arising from geometry, the study of noncommutative rings has grown into a major area of modern algebra. The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors. An incomplete list of such contributors includes
1687:
1469:
1189:
706:
to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts,
1059:
1793:. It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side. For commutative rings, the left–right distinction does not exist.
1911:
A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary.
834:
923:
1546:
1323:
235:
2772:
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
527:
1231:
1515:
966:
480:
443:
1328:
1259:
189:
2295:
of a vector space. This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings
Without Finiteness Assumptions" by
1070:
1535:
1279:
2246:
As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a
971:
3472:
1192:
718:
Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.
632:
3196:
3115:
right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right
782:
3294:
3233:
839:
3167:
1682:{\displaystyle \mathbb {C} \langle \theta _{1},\ldots ,\theta _{m}\rangle /(\theta _{i}\theta _{j}+\theta _{j}\theta _{i})}
695:
devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.
89:
3532:
2719:* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a
3540:
3458:
3157:
1900:
over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the
625:
577:
1882:. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".
3162:
1288:
199:
3267:. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.
2102:
1938:
still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an
1875:
494:
1718:
over the field with two elements; it has eight elements and all noncommutative rings with eight elements are
1198:
3101:
2146:
2028:
618:
485:
3097:
1474:
933:
335:
3147:
3127:
95:
1464:{\displaystyle \mathbb {F} \langle e_{1},\ldots ,e_{n}\rangle /(e_{i}e_{j}+e_{j}e_{i}-q(e_{i},e_{j}))}
456:
419:
110:
3075:
3071:
2621:
2606:), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma.
2583:
2024:), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns).
1715:
772:
Some examples of rings that are not typically commutative (but may be commutative in simple cases):
2300:
2267:
1955:
1789:
difference between rings which are and are not commutative is the necessity to separately consider
570:
373:
323:
2594:, and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that
1242:
172:
3152:
382:
116:
75:
3286:
3524:
2969:
2911:
2668:
2152:
2130:
1829:
1817:
1753:
539:
390:
341:
122:
968:, being the ring of polynomial differential operators defined over affine space; for example,
2977:
2781:
2292:
1901:
1184:{\displaystyle \mathbb {C} \langle x_{1},\ldots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}
3573:
2935:
2758:
2111:
1989:
1897:
658:
263:
137:
3243:
8:
3034:
2896:
2785:
2414:
2115:
2067:
1997:
1977:
1867:
353:
304:
249:
143:
129:
57:
25:
3134:
principal right ideal rings. Every prime principal right ideal ring is isomorphic to a
2814:
that preserves many ring-theoretic properties. It is named after
Japanese mathematician
3510:
3491:
3279:
3060:
2984:
2965:
2907:
2811:
2805:
2747:
2720:
2707:
2674:
2160:
1973:
1921:
1813:
1520:
1264:
654:
558:
44:
2765:
of prospective units. One condition which ensures that the localization exists is the
1926:
A semiprimitive ring or
Jacobson semisimple ring or J-semisimple ring is a ring whose
3536:
3454:
3290:
3229:
3067:
2931:
2690:
2495:
2251:
1947:
1790:
1777:
599:
396:
161:
102:
3502:
3481:
3319:
3264:
3239:
3111:, since by definition right Noetherian rings have the ascending chain condition on
2961:
2923:
2878:
2424:
2168:
2134:
1927:
1236:
1062:
682:
605:
591:
405:
347:
310:
83:
69:
2084:
An example of a simple ring that is not a matrix ring over a division ring is the
1054:{\displaystyle A_{1}(\mathbb {C} )\cong \mathbb {C} \langle x,y\rangle /(xy-yx-1)}
3557:
3467:
3256:
3225:
3120:
3108:
3090:
2939:
2743:
2571:
2296:
2156:
1943:
1931:
1891:
1761:
367:
317:
155:
3126:
A consequence of Goldie's theorem, again due to Goldie, is that every semiprime
3100:. The structure of this ring of quotients is then completely determined by the
3086:
2927:
2919:
2793:
2288:
2052:
1985:
1951:
1860:
1757:
1745:
411:
2957:
1773:
3567:
3549:
3442:
3260:
3093:
3070:(also called "finite rank") as a right module over itself, and satisfies the
3052:
2951:
2903:
2766:
2511:
2307:
2277:
2229:
2214:
2189:
2123:
2075:
2063:
2040:
2032:
1939:
1935:
1905:
1807:
1783:
1736:
1723:
766:
552:
448:
63:
3470:(1945), "Structure theory of simple rings without finiteness assumptions",
2890:
2815:
2673:
Localization is a systematic method of adding multiplicative inverses to a
2649:
2085:
2044:
1879:
1769:
1765:
1704:
1540:
928:
584:
359:
255:
3450:
3186:
3135:
3048:
3040:
2304:
2247:
2225:
2210:
2185:
2119:
2048:
2036:
1993:
1967:
1871:
743:
731:
692:
646:
564:
275:
149:
31:
3495:
3131:
3116:
2738:; however other notations are used in some important special cases. If
2255:
2205:, both of which are uniquely determined up to permutation of the index
2108:
2056:
1981:
1749:
1741:
1719:
762:
756:
329:
2043:. In particular, the only simple rings that are a finite-dimensional
1543:
are another example of noncommutative rings; they can be presented as
836:
generated by a finite set, an example of two non-equal elements being
3191:"Sequence A127708 (Number of non-commutative rings with 1)"
1821:
777:
289:
194:
3486:
2648:
A version of the lemma holds for right modules over non-commutative
1239:
can be described explicitly using an algebra presentation: given an
1859:. Stated differently, a ring is a division ring if and only if its
283:
269:
2837:
if there is an equivalence of the category of (left) modules over
2818:
who defined equivalence and a similar notion of duality in 1958.
2761:
is more difficult; the localization does not exist for every set
167:
51:
3263:. Some authors use "semisimple" to mean the ring has a trivial
829:{\displaystyle \mathbb {Z} \langle x_{1},\ldots ,x_{n}\rangle }
3021:. A domain that satisfies the right Ore condition is called a
2869:
are equivalent. Further it can be shown that any functor from
3119:
is a right Goldie domain, and hence so is every commutative
2677:, and is usually applied to commutative rings. Given a ring
3190:
3107:
In particular, Goldie's theorem applies to semiprime right
2784:. There is now a large mathematical theory about it, named
2455:
to be the set of all (finite) sums of elements of the form
2861:
are equivalent if and only if the right module categories
2000:
does not have any nontrivial ideals (since any ideal of M(
1784:
Differences between commutative and noncommutative algebra
3222:
Points and lines. Characterizing the classical geometries
2163:. The theorem states that an (Artinian) semisimple ring
2299:. This can be viewed as a kind of generalization of the
2051:
are rings of matrices over either the real numbers, the
1984:
and itself. A simple ring can always be considered as a
1878:
all finite division rings are commutative and therefore
1870:
only in that their multiplication is not required to be
1691:
There are finite noncommutative rings: for example, the
3318:
Such rings of linear transformations are also known as
2566:
need not contain any maximal submodules. Naturally, if
1325:, the associated Clifford algebra has the presentation
3185:
2960:, in connection with the question of extending beyond
1714:. The smallest noncommutative ring is the ring of the
918:{\displaystyle 2x_{1}x_{2}+x_{2}x_{1}\neq 3x_{1}x_{2}}
2810:
Morita equivalence is a relationship defined between
2313:
More formally, the theorem can be stated as follows:
1549:
1523:
1477:
1331:
1291:
1267:
1245:
1201:
1073:
974:
936:
842:
785:
497:
459:
422:
202:
175:
2554:. However, this need not hold for arbitrary modules
2258:
later generalized it to the case of
Artinian rings.
1930:
is zero. This is a type of ring more general than a
3089:right Goldie rings are precisely those that have a
2655:. The resulting theorem is sometimes known as the
2137:: every finite simple alternative ring is a field.
3278:
2922:of algebras. It arose out of attempts to classify
2853:. It can be shown that the left module categories
2542:contains at least one (proper) maximal submodule,
2291:can be viewed as a "dense" subring of the ring of
1681:
1529:
1509:
1463:
1317:
1273:
1253:
1225:
1183:
1053:
960:
917:
828:
521:
474:
437:
229:
183:
3473:Transactions of the American Mathematical Society
3565:
1812:A division ring, also called a skew field, is a
2956:The Ore condition is a condition introduced by
2926:over a field and is named after the algebraist
2788:, connecting with numerous other branches. The
2780:. This is done in many contexts in methods for
3276:
2662:
2107:Wedderburn's little theorem states that every
2096:
1988:. Rings which are simple as rings but not as
3055:during the 1950s. What is now termed a right
2930:. The group may also be defined in terms of
626:
2877:that yields an equivalence is automatically
2287:The theorem can be applied to show that any
1946:. Semiprimitive rings can be understood as
1622:
1590:
1369:
1337:
1318:{\displaystyle q:V\otimes V\to \mathbb {F} }
1111:
1079:
1016:
1004:
823:
791:
230:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
2261:
2140:
2122:, there is no distinction between domains,
3428:. Vol. 3 (2nd ed.). p. 351.
3281:Introductory Lectures on Rings and Modules
2845:, and the category of (left) modules over
2742:is the set of the non zero elements of an
2031:, every simple ring that is left or right
633:
619:
3485:
3395:
3393:
3270:
3197:On-Line Encyclopedia of Integer Sequences
2934:. More generally, the Brauer group of a
2602:-module (and no finiteness assumption on
1551:
1333:
1311:
1247:
1219:
1075:
1000:
989:
951:
787:
522:{\displaystyle \mathbb {Z} (p^{\infty })}
499:
462:
425:
223:
210:
177:
3466:
3340:
3325:
3130:is isomorphic to a finite direct sum of
726:Some examples of noncommutative rings:
3566:
3554:A First Course in Noncommutative Rings
3501:
3441:
3411:
3399:
3390:
3384:
3372:
3360:
3285:. Cambridge University Press. p.
3025:. The left case is defined similarly.
2829:(associative, with 1) are said to be (
1915:
1226:{\displaystyle q_{ij}\in \mathbb {C} }
3219:
2799:
2303:'s conclusion about the structure of
2091:
1061:, where the ideal corresponds to the
765:constructed from a group that is not
3423:
3168:Representation theory (group theory)
1863:is the set of all nonzero elements.
1824:ring in which every nonzero element
1796:
90:Free product of associative algebras
3533:Mathematical Association of America
3028:
2792:tag is to do with connections with
2685:, one wants to construct some ring
2408:
2066:is a simple ring. In particular, a
1885:
1820:is possible. Specifically, it is a
1510:{\displaystyle e_{1},\ldots ,e_{n}}
961:{\displaystyle A_{n}(\mathbb {C} )}
13:
3517:
2609:Precisely, one has the following.
2151:The Artin–Wedderburn theorem is a
511:
14:
3585:
3424:Cohn, P. M. (1991). "Chap. 9.1".
3158:Noncommutative algebraic geometry
3085:Goldie's theorem states that the
2945:
1801:
578:Noncommutative algebraic geometry
3210:In this article, rings have a 1.
3163:Noncommutative harmonic analysis
3047:is a basic structural result in
475:{\displaystyle \mathbb {Q} _{p}}
438:{\displaystyle \mathbb {Z} _{p}}
3417:
3405:
3378:
3366:
3354:
3345:
3331:Isaacs, Corollary 13.16, p. 187
2918:and addition is induced by the
2884:
2776:for a differentiation operator
2746:, then the localization is the
2435:is a right module over a ring,
1961:
677:are different. Equivalently, a
3334:
3312:
3303:
3250:
3213:
3204:
3179:
2750:and thus usually denoted Frac(
2356:-linearly independent set. If
1676:
1630:
1587:
1555:
1458:
1455:
1429:
1377:
1307:
1178:
1119:
1048:
1024:
993:
985:
955:
947:
516:
503:
1:
3435:
3351:Isaacs, Theorem 13.14, p. 185
2522:) is necessarily a subset of
2318:The Jacobson Density Theorem.
2074:is simple if and only if its
1954:, which are described by the
1942:semiprimitive ring is just a
2598:is finitely generated as an
2590:is a Noetherian module over
2062:Any quotient of a ring by a
1972:A simple ring is a non-zero
1791:right ideals and left ideals
1254:{\displaystyle \mathbb {F} }
657:whose multiplication is not
184:{\displaystyle \mathbb {Z} }
7:
3141:
3098:classical ring of quotients
2663:Noncommutative localization
2640:) is a proper submodule of
2632:is a non-zero module, then
2550:) is a proper submodule of
2133:generalizes the theorem to
2103:Wedderburn's little theorem
2097:Wedderburn's little theorem
1876:Wedderburn's little theorem
1866:Division rings differ from
721:
711:is used as a shorthand for
336:Unique factorization domain
10:
3590:
3447:Algebra, a graduate course
3187:Sloane, N. J. A.
3148:Derived algebraic geometry
3128:principal right ideal ring
3032:
2949:
2888:
2803:
2701:*, such that the image of
2666:
2412:
2364:-linear transformation on
2265:
2144:
2100:
1965:
1919:
1908:(irreducible) submodules.
1889:
1805:
1730:
96:Tensor product of algebras
3220:Shult, Ernest E. (2011).
3138:over a right Ore domain.
3072:ascending chain condition
2711:(invertible elements) in
2624:right module over a ring
2526:, by the definition of J(
2243:are uniquely determined.
2070:is a simple ring. A ring
1716:upper triangular matrices
3224:. Universitext. Berlin:
3173:
3102:Artin–Wedderburn theorem
2657:Jacobson–Azumaya theorem
2467:is simply the action of
2301:Artin-Wedderburn theorem
2276:is a theorem concerning
2274:Jacobson density theorem
2268:Jacobson density theorem
2262:Jacobson density theorem
2147:Artin–Wedderburn theorem
2141:Artin–Wedderburn theorem
2029:Artin–Wedderburn theorem
1956:Jacobson density theorem
681:is a ring that is not a
374:Formal power series ring
324:Integrally closed domain
3402:, Theorem 13.11, p. 183
3387:, Theorem 12.19, p. 172
3277:John A. Beachy (1999).
3153:Noncommutative geometry
2938:is defined in terms of
2912:central simple algebras
2574:module, this holds. If
661:; that is, there exist
383:Algebraic number theory
76:Total ring of fractions
2970:localization of a ring
2964:the construction of a
2895:The Brauer group of a
2782:differential equations
2731:is usually denoted by
2723:. The localization of
2669:Localization of a ring
2293:linear transformations
2209:. In particular, any
2153:classification theorem
2118:. In other words, for
1976:that has no two-sided
1830:multiplicative inverse
1683:
1531:
1511:
1465:
1319:
1285:with a quadratic form
1275:
1255:
1227:
1185:
1055:
962:
919:
830:
689:Noncommutative algebra
669:in the ring such that
540:Noncommutative algebra
523:
476:
439:
391:Algebraic number field
342:Principal ideal domain
231:
185:
123:Frobenius endomorphism
2978:multiplicative subset
2759:non-commutative rings
2715:*. Further one wants
2254:'s original result.
1684:
1532:
1512:
1466:
1320:
1276:
1256:
1228:
1186:
1056:
963:
920:
831:
524:
477:
440:
232:
186:
3529:Noncommutative Rings
2968:, or more generally
2914:of finite rank over
2538:is simple. Thus, if
2530:) and the fact that
2443:is a right ideal in
2217:is isomorphic to an
2198:, for some integers
1547:
1521:
1475:
1329:
1289:
1265:
1243:
1199:
1071:
972:
934:
840:
783:
546:Noncommutative rings
495:
457:
420:
264:Non-associative ring
200:
173:
130:Algebraic structures
3010:, the intersection
2974:right Ore condition
2906:whose elements are
2578:is Noetherian, and
2167:is isomorphic to a
2161:semisimple algebras
2008:) is of the form M(
1992:do exist: the full
1916:Semiprimitive rings
1832:, i.e., an element
702:is used instead of
700:noncommutative ring
698:Sometimes the term
679:noncommutative ring
651:noncommutative ring
305:Commutative algebra
144:Associative algebra
26:Algebraic structure
16:Algebraic structure
3511:Wiley-Interscience
3200:. OEIS Foundation.
2966:field of fractions
2908:Morita equivalence
2806:Morita equivalence
2800:Morita equivalence
2748:field of fractions
2721:universal property
2622:finitely generated
2584:finitely generated
2483:is a submodule of
2368:then there exists
2324:be a simple right
2131:Artin–Zorn theorem
2092:Important theorems
1948:subdirect products
1922:Semiprimitive ring
1679:
1527:
1507:
1461:
1315:
1271:
1251:
1223:
1181:
1067:The quotient ring
1051:
958:
915:
826:
742:matrices over the
559:Semiprimitive ring
519:
472:
435:
243:Related structures
227:
181:
117:Inner automorphism
103:Ring homomorphisms
3443:Isaacs, I. Martin
3320:full linear rings
3296:978-0-521-64407-5
3235:978-3-642-15626-7
3068:uniform dimension
2962:commutative rings
2932:Galois cohomology
2924:division algebras
2796:, in particular.
2786:microlocalization
2691:ring homomorphism
2496:maximal submodule
2252:Joseph Wedderburn
2171:of finitely many
2135:alternative rings
2027:According to the
1797:Important classes
1530:{\displaystyle V}
1274:{\displaystyle V}
643:
642:
600:Geometric algebra
311:Commutative rings
162:Category of rings
3581:
3560:
3545:
3513:
3498:
3489:
3463:
3449:(1st ed.),
3430:
3429:
3421:
3415:
3409:
3403:
3397:
3388:
3382:
3376:
3370:
3364:
3358:
3352:
3349:
3343:
3338:
3332:
3329:
3323:
3316:
3310:
3307:
3301:
3300:
3284:
3274:
3268:
3265:Jacobson radical
3259:are necessarily
3257:Semisimple rings
3254:
3248:
3247:
3217:
3211:
3208:
3202:
3201:
3183:
3109:Noetherian rings
3066:that has finite
3045:Goldie's theorem
3035:Goldie's theorem
3029:Goldie's theorem
3023:right Ore domain
3020:
3009:
2999:
2940:Azumaya algebras
2614:Nakayama's lemma
2425:Jacobson radical
2415:Nakayama's lemma
2409:Nakayama's lemma
2403:
2399:
2395:
2377:
2367:
2363:
2359:
2355:
2351:
2341:
2327:
2323:
2283:
2157:semisimple rings
1928:Jacobson radical
1886:Semisimple rings
1858:
1722:to it or to its
1713:
1703:matrices over a
1702:
1696:
1688:
1686:
1685:
1680:
1675:
1674:
1665:
1664:
1652:
1651:
1642:
1641:
1629:
1621:
1620:
1602:
1601:
1586:
1585:
1567:
1566:
1554:
1536:
1534:
1533:
1528:
1516:
1514:
1513:
1508:
1506:
1505:
1487:
1486:
1470:
1468:
1467:
1462:
1454:
1453:
1441:
1440:
1422:
1421:
1412:
1411:
1399:
1398:
1389:
1388:
1376:
1368:
1367:
1349:
1348:
1336:
1324:
1322:
1321:
1316:
1314:
1284:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1250:
1237:Clifford algebra
1232:
1230:
1229:
1224:
1222:
1214:
1213:
1190:
1188:
1187:
1182:
1177:
1176:
1167:
1166:
1157:
1156:
1141:
1140:
1131:
1130:
1118:
1110:
1109:
1091:
1090:
1078:
1060:
1058:
1057:
1052:
1023:
1003:
992:
984:
983:
967:
965:
964:
959:
954:
946:
945:
924:
922:
921:
916:
914:
913:
904:
903:
888:
887:
878:
877:
865:
864:
855:
854:
835:
833:
832:
827:
822:
821:
803:
802:
790:
752:
713:commutative ring
683:commutative ring
635:
628:
621:
606:Operator algebra
592:Clifford algebra
528:
526:
525:
520:
515:
514:
502:
481:
479:
478:
473:
471:
470:
465:
444:
442:
441:
436:
434:
433:
428:
406:Ring of integers
400:
397:Integers modulo
348:Euclidean domain
236:
234:
233:
228:
226:
218:
213:
190:
188:
187:
182:
180:
84:Product of rings
70:Fractional ideal
29:
21:
20:
3589:
3588:
3584:
3583:
3582:
3580:
3579:
3578:
3564:
3563:
3558:Springer-Verlag
3548:
3543:
3525:Herstein, I. N.
3523:
3520:
3518:Further reading
3487:10.2307/1990204
3461:
3438:
3433:
3422:
3418:
3410:
3406:
3398:
3391:
3383:
3379:
3371:
3367:
3359:
3355:
3350:
3346:
3339:
3335:
3330:
3326:
3317:
3313:
3308:
3304:
3297:
3275:
3271:
3255:
3251:
3236:
3228:. p. 123.
3226:Springer-Verlag
3218:
3214:
3209:
3205:
3184:
3180:
3176:
3144:
3121:integral domain
3037:
3031:
3011:
3001:
2991:
2954:
2948:
2893:
2887:
2808:
2802:
2744:integral domain
2671:
2665:
2475:. Necessarily,
2417:
2411:
2401:
2397:
2379:
2369:
2365:
2361:
2357:
2353:
2343:
2338:
2329:
2325:
2321:
2297:Nathan Jacobson
2281:
2270:
2264:
2203:
2196:
2183:
2176:
2149:
2143:
2105:
2099:
2094:
2053:complex numbers
1970:
1964:
1952:primitive rings
1944:semisimple ring
1932:semisimple ring
1924:
1918:
1894:
1892:Semisimple ring
1888:
1837:
1810:
1804:
1799:
1786:
1735:Beginning with
1733:
1708:
1698:
1692:
1670:
1666:
1660:
1656:
1647:
1643:
1637:
1633:
1625:
1616:
1612:
1597:
1593:
1581:
1577:
1562:
1558:
1550:
1548:
1545:
1544:
1522:
1519:
1518:
1501:
1497:
1482:
1478:
1476:
1473:
1472:
1449:
1445:
1436:
1432:
1417:
1413:
1407:
1403:
1394:
1390:
1384:
1380:
1372:
1363:
1359:
1344:
1340:
1332:
1330:
1327:
1326:
1310:
1290:
1287:
1286:
1282:
1266:
1263:
1262:
1246:
1244:
1241:
1240:
1218:
1206:
1202:
1200:
1197:
1196:
1172:
1168:
1162:
1158:
1149:
1145:
1136:
1132:
1126:
1122:
1114:
1105:
1101:
1086:
1082:
1074:
1072:
1069:
1068:
1019:
999:
988:
979:
975:
973:
970:
969:
950:
941:
937:
935:
932:
931:
909:
905:
899:
895:
883:
879:
873:
869:
860:
856:
850:
846:
841:
838:
837:
817:
813:
798:
794:
786:
784:
781:
780:
747:
724:
691:is the part of
639:
610:
609:
542:
532:
531:
510:
506:
498:
496:
493:
492:
466:
461:
460:
458:
455:
454:
429:
424:
423:
421:
418:
417:
398:
368:Polynomial ring
318:Integral domain
307:
297:
296:
222:
214:
209:
201:
198:
197:
176:
174:
171:
170:
156:Involutive ring
41:
30:
24:
17:
12:
11:
5:
3587:
3577:
3576:
3562:
3561:
3546:
3541:
3519:
3516:
3515:
3514:
3499:
3464:
3459:
3437:
3434:
3432:
3431:
3416:
3404:
3389:
3377:
3365:
3353:
3344:
3333:
3324:
3311:
3309:Isaacs, p. 184
3302:
3295:
3269:
3261:Artinian rings
3249:
3234:
3212:
3203:
3177:
3175:
3172:
3171:
3170:
3165:
3160:
3155:
3150:
3143:
3140:
3078:of subsets of
3033:Main article:
3030:
3027:
2950:Main article:
2947:
2946:Ore conditions
2944:
2928:Richard Brauer
2920:tensor product
2889:Main article:
2886:
2883:
2804:Main article:
2801:
2798:
2794:Fourier theory
2667:Main article:
2664:
2661:
2646:
2645:
2447:, then define
2413:Main article:
2410:
2407:
2406:
2405:
2336:
2308:Artinian rings
2289:primitive ring
2278:simple modules
2266:Main article:
2263:
2260:
2213:left or right
2201:
2194:
2190:division rings
2181:
2174:
2145:Main article:
2142:
2139:
2124:division rings
2101:Main article:
2098:
2095:
2093:
2090:
1986:simple algebra
1966:Main article:
1963:
1960:
1936:simple modules
1920:Main article:
1917:
1914:
1890:Main article:
1887:
1884:
1874:. However, by
1861:group of units
1806:Main article:
1803:
1802:Division rings
1800:
1798:
1795:
1785:
1782:
1758:I. N. Herstein
1754:W. R. Hamilton
1746:Richard Brauer
1737:division rings
1732:
1729:
1728:
1727:
1689:
1678:
1673:
1669:
1663:
1659:
1655:
1650:
1646:
1640:
1636:
1632:
1628:
1624:
1619:
1615:
1611:
1608:
1605:
1600:
1596:
1592:
1589:
1584:
1580:
1576:
1573:
1570:
1565:
1561:
1557:
1553:
1538:
1526:
1504:
1500:
1496:
1493:
1490:
1485:
1481:
1471:for any basis
1460:
1457:
1452:
1448:
1444:
1439:
1435:
1431:
1428:
1425:
1420:
1416:
1410:
1406:
1402:
1397:
1393:
1387:
1383:
1379:
1375:
1371:
1366:
1362:
1358:
1355:
1352:
1347:
1343:
1339:
1335:
1313:
1309:
1306:
1303:
1300:
1297:
1294:
1270:
1261:-vector space
1249:
1233:
1221:
1217:
1212:
1209:
1205:
1180:
1175:
1171:
1165:
1161:
1155:
1152:
1148:
1144:
1139:
1135:
1129:
1125:
1121:
1117:
1113:
1108:
1104:
1100:
1097:
1094:
1089:
1085:
1081:
1077:
1065:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1022:
1018:
1015:
1012:
1009:
1006:
1002:
998:
995:
991:
987:
982:
978:
957:
953:
949:
944:
940:
925:
912:
908:
902:
898:
894:
891:
886:
882:
876:
872:
868:
863:
859:
853:
849:
845:
825:
820:
816:
812:
809:
806:
801:
797:
793:
789:
770:
769:
759:
753:
723:
720:
641:
640:
638:
637:
630:
623:
615:
612:
611:
603:
602:
574:
573:
567:
561:
555:
543:
538:
537:
534:
533:
530:
529:
518:
513:
509:
505:
501:
482:
469:
464:
445:
432:
427:
415:-adic integers
408:
402:
393:
379:
378:
377:
376:
370:
364:
363:
362:
350:
344:
338:
332:
326:
308:
303:
302:
299:
298:
295:
294:
293:
292:
280:
279:
278:
272:
260:
259:
258:
240:
239:
238:
237:
225:
221:
217:
212:
208:
205:
191:
179:
158:
152:
146:
140:
126:
125:
119:
113:
99:
98:
92:
86:
80:
79:
78:
72:
60:
54:
42:
40:Basic concepts
39:
38:
35:
34:
15:
9:
6:
4:
3:
2:
3586:
3575:
3572:
3571:
3569:
3559:
3555:
3551:
3547:
3544:
3542:0-88385-015-X
3538:
3534:
3530:
3526:
3522:
3521:
3512:
3508:
3504:
3500:
3497:
3493:
3488:
3483:
3479:
3475:
3474:
3469:
3465:
3462:
3460:0-534-19002-2
3456:
3452:
3448:
3444:
3440:
3439:
3427:
3420:
3413:
3408:
3401:
3396:
3394:
3386:
3381:
3375:, p. 183
3374:
3369:
3363:, p. 182
3362:
3357:
3348:
3342:
3341:Jacobson 1945
3337:
3328:
3321:
3315:
3306:
3298:
3292:
3288:
3283:
3282:
3273:
3266:
3262:
3258:
3253:
3245:
3241:
3237:
3231:
3227:
3223:
3216:
3207:
3199:
3198:
3192:
3188:
3182:
3178:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3146:
3145:
3139:
3137:
3133:
3129:
3124:
3122:
3118:
3114:
3110:
3105:
3103:
3099:
3095:
3092:
3088:
3083:
3081:
3077:
3073:
3069:
3065:
3062:
3058:
3054:
3053:Alfred Goldie
3050:
3046:
3042:
3036:
3026:
3024:
3018:
3014:
3008:
3004:
2998:
2994:
2989:
2986:
2982:
2979:
2975:
2971:
2967:
2963:
2959:
2953:
2952:Ore condition
2943:
2941:
2937:
2933:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2904:abelian group
2901:
2898:
2892:
2882:
2880:
2876:
2872:
2868:
2864:
2860:
2856:
2852:
2848:
2844:
2840:
2836:
2832:
2828:
2824:
2819:
2817:
2813:
2807:
2797:
2795:
2791:
2787:
2783:
2779:
2775:
2770:
2768:
2767:Ore condition
2764:
2760:
2755:
2753:
2749:
2745:
2741:
2737:
2734:
2730:
2726:
2722:
2718:
2714:
2710:
2709:
2704:
2700:
2696:
2692:
2688:
2684:
2681:and a subset
2680:
2676:
2670:
2660:
2658:
2654:
2651:
2650:unitary rings
2643:
2639:
2635:
2631:
2627:
2623:
2619:
2615:
2612:
2611:
2610:
2607:
2605:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2553:
2549:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2488:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2434:
2430:
2426:
2422:
2416:
2394:
2390:
2386:
2382:
2376:
2372:
2352:a finite and
2350:
2346:
2339:
2332:
2319:
2316:
2315:
2314:
2311:
2309:
2306:
2302:
2298:
2294:
2290:
2285:
2279:
2275:
2269:
2259:
2257:
2253:
2249:
2244:
2242:
2238:
2235:, where both
2234:
2231:
2230:division ring
2227:
2224:
2220:
2216:
2215:Artinian ring
2212:
2208:
2204:
2197:
2191:
2187:
2184:
2177:
2170:
2166:
2162:
2158:
2154:
2148:
2138:
2136:
2132:
2127:
2125:
2121:
2117:
2113:
2110:
2104:
2089:
2087:
2082:
2080:
2077:
2076:opposite ring
2073:
2069:
2065:
2064:maximal ideal
2060:
2058:
2054:
2050:
2046:
2042:
2041:division ring
2038:
2034:
2030:
2025:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1969:
1959:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1923:
1913:
1909:
1907:
1903:
1899:
1893:
1883:
1881:
1880:finite fields
1877:
1873:
1869:
1864:
1862:
1856:
1853:
1850:
1846:
1843:
1840:
1835:
1831:
1827:
1823:
1819:
1815:
1809:
1808:Division ring
1794:
1792:
1781:
1779:
1778:J. Wedderburn
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1738:
1725:
1721:
1717:
1711:
1706:
1701:
1695:
1690:
1671:
1667:
1661:
1657:
1653:
1648:
1644:
1638:
1634:
1626:
1617:
1613:
1609:
1606:
1603:
1598:
1594:
1582:
1578:
1574:
1571:
1568:
1563:
1559:
1542:
1541:Superalgebras
1539:
1524:
1502:
1498:
1494:
1491:
1488:
1483:
1479:
1450:
1446:
1442:
1437:
1433:
1426:
1423:
1418:
1414:
1408:
1404:
1400:
1395:
1391:
1385:
1381:
1373:
1364:
1360:
1356:
1353:
1350:
1345:
1341:
1304:
1301:
1298:
1295:
1292:
1281:of dimension
1268:
1238:
1234:
1215:
1210:
1207:
1203:
1194:
1193:quantum plane
1173:
1169:
1163:
1159:
1153:
1150:
1146:
1142:
1137:
1133:
1127:
1123:
1115:
1106:
1102:
1098:
1095:
1092:
1087:
1083:
1066:
1064:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1020:
1013:
1010:
1007:
996:
980:
976:
942:
938:
930:
926:
910:
906:
900:
896:
892:
889:
884:
880:
874:
870:
866:
861:
857:
851:
847:
843:
818:
814:
810:
807:
804:
799:
795:
779:
775:
774:
773:
768:
764:
760:
758:
754:
750:
745:
741:
737:
733:
729:
728:
727:
719:
716:
714:
710:
705:
701:
696:
694:
690:
686:
684:
680:
676:
672:
668:
664:
660:
656:
652:
648:
636:
631:
629:
624:
622:
617:
616:
614:
613:
608:
607:
601:
597:
596:
595:
594:
593:
588:
587:
586:
581:
580:
579:
572:
568:
566:
562:
560:
556:
554:
553:Division ring
550:
549:
548:
547:
541:
536:
535:
507:
491:
489:
483:
467:
453:
452:-adic numbers
451:
446:
430:
416:
414:
409:
407:
403:
401:
394:
392:
388:
387:
386:
385:
384:
375:
371:
369:
365:
361:
357:
356:
355:
351:
349:
345:
343:
339:
337:
333:
331:
327:
325:
321:
320:
319:
315:
314:
313:
312:
306:
301:
300:
291:
287:
286:
285:
281:
277:
273:
271:
267:
266:
265:
261:
257:
253:
252:
251:
247:
246:
245:
244:
219:
215:
206:
203:
196:
195:Terminal ring
192:
169:
165:
164:
163:
159:
157:
153:
151:
147:
145:
141:
139:
135:
134:
133:
132:
131:
124:
120:
118:
114:
112:
108:
107:
106:
105:
104:
97:
93:
91:
87:
85:
81:
77:
73:
71:
67:
66:
65:
64:Quotient ring
61:
59:
55:
53:
49:
48:
47:
46:
37:
36:
33:
28:→ Ring theory
27:
23:
22:
19:
3553:
3528:
3506:
3477:
3471:
3468:Jacobson, N.
3446:
3425:
3419:
3407:
3380:
3368:
3356:
3347:
3336:
3327:
3314:
3305:
3280:
3272:
3252:
3221:
3215:
3206:
3194:
3181:
3125:
3112:
3106:
3084:
3079:
3076:annihilators
3063:
3056:
3051:, proved by
3044:
3038:
3022:
3016:
3012:
3006:
3002:
2996:
2992:
2990:is that for
2987:
2980:
2973:
2955:
2915:
2899:
2894:
2891:Brauer group
2885:Brauer group
2874:
2870:
2866:
2862:
2858:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2820:
2816:Kiiti Morita
2809:
2789:
2777:
2773:
2771:
2762:
2756:
2751:
2739:
2735:
2732:
2728:
2724:
2716:
2712:
2706:
2705:consists of
2702:
2698:
2694:
2686:
2682:
2678:
2672:
2656:
2652:
2647:
2641:
2637:
2633:
2629:
2625:
2617:
2613:
2608:
2603:
2599:
2595:
2591:
2587:
2579:
2575:
2567:
2563:
2559:
2555:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2519:
2515:
2507:
2503:
2499:
2491:
2489:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2440:
2436:
2432:
2428:
2420:
2418:
2392:
2388:
2384:
2380:
2374:
2370:
2348:
2344:
2334:
2330:
2317:
2312:
2286:
2280:over a ring
2273:
2271:
2245:
2240:
2236:
2232:
2222:
2218:
2206:
2199:
2192:
2186:matrix rings
2179:
2172:
2164:
2150:
2128:
2126:and fields.
2120:finite rings
2106:
2086:Weyl algebra
2083:
2078:
2071:
2061:
2049:real numbers
2045:vector space
2026:
2021:
2020:an ideal of
2017:
2013:
2009:
2005:
2001:
1980:besides the
1971:
1962:Simple rings
1934:, but where
1925:
1910:
1895:
1865:
1854:
1851:
1848:
1844:
1841:
1838:
1833:
1825:
1811:
1787:
1780:and others.
1734:
1709:
1705:finite field
1699:
1693:
929:Weyl algebra
771:
748:
744:real numbers
739:
735:
725:
717:
712:
708:
703:
699:
697:
688:
687:
678:
674:
670:
666:
662:
650:
644:
604:
590:
589:
585:Free algebra
583:
582:
576:
575:
545:
544:
487:
449:
412:
381:
380:
360:Finite field
309:
256:Finite field
242:
241:
168:Initial ring
128:
127:
101:
100:
43:
18:
3574:Ring theory
3507:Local Rings
3480:: 228–245,
3451:Brooks/Cole
3412:Nagata 1962
3400:Isaacs 1993
3385:Isaacs 1993
3373:Isaacs 1993
3361:Isaacs 1993
3136:matrix ring
3057:Goldie ring
3049:ring theory
3041:mathematics
2958:Øystein Ore
2910:classes of
2757:Localizing
2250:. This is
2248:matrix ring
2226:matrix ring
2081:is simple.
2057:quaternions
2037:matrix ring
1994:matrix ring
1968:Simple ring
1872:commutative
1762:N. Jacobson
1191:, called a
757:quaternions
755:Hamilton's
732:matrix ring
693:ring theory
659:commutative
647:mathematics
565:Simple ring
276:Jordan ring
150:Graded ring
32:Ring theory
3550:Lam, T. Y.
3503:Nagata, M.
3436:References
3244:1213.51001
3117:Ore domain
3091:semisimple
2835:equivalent
2821:Two rings
2572:Noetherian
2378:such that
2256:Emil Artin
1982:zero ideal
1902:direct sum
1770:E. Noether
1750:P. M. Cohn
1720:isomorphic
1063:commutator
763:group ring
571:Commutator
330:GCD domain
3087:semiprime
3074:on right
2423:) be the
2328:-module,
2055:, or the
2047:over the
1816:in which
1766:K. Morita
1668:θ
1658:θ
1645:θ
1635:θ
1623:⟩
1614:θ
1607:…
1595:θ
1591:⟨
1572:…
1492:…
1424:−
1370:⟩
1354:…
1338:⟨
1308:→
1302:⊗
1216:∈
1143:−
1112:⟩
1096:…
1080:⟨
1043:−
1034:−
1017:⟩
1005:⟨
997:≅
890:≠
824:⟩
808:…
792:⟨
778:free ring
512:∞
290:Semifield
3568:Category
3552:(2001),
3527:(1968),
3505:(1962),
3445:(1993),
3142:See also
3094:Artinian
2879:additive
2463:, where
2396:for all
2033:Artinian
1940:artinian
1818:division
1742:E. Artin
1724:opposite
1195:, where
746:, where
722:Examples
284:Semiring
270:Lie ring
52:Subrings
3496:1990204
3426:Algebra
3189:(ed.).
2586:, then
2502:, then
2228:over a
2169:product
2039:over a
2016:) with
1996:over a
1990:modules
1822:nonzero
1731:History
767:abelian
486:Prüfer
88:•
3539:
3494:
3457:
3293:
3242:
3232:
3096:right
2976:for a
2972:. The
2936:scheme
2902:is an
2831:Morita
2790:micro-
2689:* and
2628:. If
2616:: Let
2562:, for
2514:. So
2512:simple
2439:, and
2419:Let J(
2342:, and
2333:= End(
2305:simple
2211:simple
2112:domain
2109:finite
1906:simple
1898:module
1868:fields
1828:has a
1774:Ø. Ore
1712:> 1
1707:, for
751:> 1
138:Module
111:Kernel
3492:JSTOR
3414:, §A2
3174:Notes
3132:prime
3059:is a
2983:of a
2897:field
2875:S-Mod
2871:R-Mod
2867:Mod-S
2863:Mod-R
2859:S-Mod
2855:R-Mod
2851:S-Mod
2843:R-Mod
2812:rings
2708:units
2693:from
2620:be a
2570:is a
2558:over
2494:is a
2431:. If
2360:is a
2188:over
2116:field
2114:is a
2068:field
2035:is a
1998:field
1978:ideal
1836:with
653:is a
490:-ring
354:Field
250:Field
58:Ideal
45:Rings
3537:ISBN
3455:ISBN
3291:ISBN
3230:ISBN
3195:The
3061:ring
3000:and
2985:ring
2865:and
2857:and
2825:and
2675:ring
2387:) =
2320:Let
2272:The
2239:and
2221:-by-
2178:-by-
2159:and
2155:for
2129:The
1974:ring
1814:ring
1697:-by-
1235:Any
927:The
776:The
761:Any
738:-by-
730:The
709:ring
704:ring
673:and
665:and
655:ring
649:, a
3482:doi
3287:156
3240:Zbl
3113:all
3039:In
3019:≠ ∅
2873:to
2754:).
2727:by
2697:to
2636:·J(
2582:is
2546:·J(
2518:·J(
2510:is
2498:of
2490:If
2471:on
2427:of
2400:in
1950:of
1904:of
1857:= 1
1517:of
734:of
645:In
3570::
3556:,
3535:,
3531:,
3509:,
3490:,
3478:57
3476:,
3453:,
3392:^
3289:.
3238:.
3193:.
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3104:.
3082:.
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3017:sR
3015:∩
3013:aS
3005:∈
2995:∈
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2881:.
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2841:,
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2769:.
2659:.
2487:.
2391:·
2373:∈
2347:⊂
2310:.
2284:.
2088:.
2059:.
1958:.
1896:A
1847:=
1776:,
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715:.
685:.
675:ba
671:ab
598:•
569:•
563:•
557:•
551:•
484:•
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404:•
395:•
389:•
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340:•
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288:•
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3484::
3322:.
3299:.
3246:.
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3064:R
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2997:R
2993:a
2988:R
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2916:K
2900:K
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2827:S
2823:R
2778:D
2774:D
2763:S
2752:R
2740:S
2736:R
2733:S
2729:S
2725:R
2717:R
2713:R
2703:S
2699:R
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2687:R
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2644:.
2642:U
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2634:U
2630:U
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2532:U
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2079:R
2072:R
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2018:I
2014:I
2012:,
2010:n
2006:R
2004:,
2002:n
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504:(
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463:Q
450:p
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413:p
399:n
224:Z
220:1
216:/
211:Z
207:=
204:0
178:Z
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