418:
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, and one can speak of semisimple rings without ambiguity.
1074:
is zero. Every ring that is semisimple as a module over itself has zero
Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself. A J-semisimple ring is semisimple if and only if it is an
1029:
1038:. These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings. The
497:
630:
571:
1342:
951:
1292:
1233:
1199:
1178:
1266:
444:
1318:
84:
is initially understood via its largest semisimple quotient. The structure of
Artinian semisimple rings is well understood by the
1280:
1213:
785:
85:
778:
346:
196:
101:
762:
644:
402:
582:
1360:
232:
228:
224:
126:
544:
1365:
1035:
774:
431:
114:
1099:
423:
350:
77:
73:
1336:
1104:
1066:
1051:
926:. The problem is that the ring may be "too big", that is, not (left/right) Artinian. In fact, if
700:
391:
57:
31:
1042:
for the Weyl algebras is well studied and differs significantly from that of semisimple rings.
1324:
1314:
1288:
1262:
1229:
1195:
1174:
640:
398:
372:
1306:
1221:
1071:
713:
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686:
636:
41:
1243:
1284:
1258:
1239:
1217:
1187:
1170:
736:
732:
717:
682:
650:
Semisimple rings are of particular interest to algebraists. For example, if the base ring
685:. From the above properties, a ring is semisimple if and only if it is Artinian and its
161:
89:
1354:
1076:
1039:
858:
721:
678:
380:
130:
81:
45:
938:
729:
725:
383:
204:
69:
1310:
917:
203:
The most basic example of a semisimple module is a module over a field, i.e., a
93:
37:
17:
1250:
1225:
1024:{\displaystyle A=\mathbf {Q} {\langle x,y\rangle }/\langle xy-yx-1\rangle \ ,}
755:
671:
335:
65:
1328:
658:-modules would automatically be semisimple. Furthermore, every simple (left)
314:
1070:) if the intersection of the maximal left ideals is zero, that is, if the
213:
of integers is not a semisimple module over itself, since the submodule
696:
56:
is a type of module that can be understood easily from its parts. A
474:
461:
903:), the row-finite, column-finite, infinite matrices over a field
60:
that is a semisimple module over itself is known as an
Artinian
492:{\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0}
937:
Classic examples of simple, but not semisimple, rings are the
394:, and every semiprimitive ring is isomorphic to such an image.
739:
0; and being isomorphic to a finite direct product of fields.
197:
Semisimple representation § Equivalent characterizations
643:. Since "projective" implies "flat", a semisimple ring is a
401:
of a semisimple module is not only semiprimitive, but also
147:
is semisimple; i.e., a direct sum of irreducible modules.
716:, the four following properties are equivalent: being a
117:
over a (not necessarily commutative) ring is said to be
930:
is a simple ring with a minimal left/right ideal, then
1303:
Representing finite groups: a semisimple introduction
954:
692:
If an
Artinian semisimple ring contains a field as a
585:
547:
447:
438:-modules splits. That is, for a short exact sequence
349:
and semisimple if and only if it is
Artinian and its
635:
In particular, any module over a semisimple ring is
1301:Sengupta, Ambar (2012). "Induced Representations".
792:is semisimple if and only if it is (isomorphic to)
422:A semisimple ring may be characterized in terms of
1216:, vol. 131 (2nd ed.), Berlin, New York:
1023:
624:
565:
491:
99:For a group-theory analog of the same notion, see
662:-module is isomorphic to a minimal left ideal of
538:is known as a section. From this it follows that
1090:, is J-semisimple, but not artinian semisimple.
922:One should beware that despite the terminology,
895:An example of a semisimple non-unital ring is M
8:
1012:
988:
979:
967:
1341:: CS1 maint: location missing publisher (
983:
966:
961:
953:
584:
546:
446:
153:is the sum of its irreducible submodules.
1147:
1135:
1116:
1079:, so semisimple rings are often called
625:{\displaystyle B\cong f(A)\oplus s(C).}
390:. The image of this homomorphism is a
194:For the proof of the equivalences, see
88:, which exhibits these rings as finite
1334:
1210:A First Course in Noncommutative Rings
7:
338:of semisimple modules is semisimple.
1123:
1086:For example, the ring of integers,
924:not all simple rings are semisimple
1257:(3rd ed.), Berlin, New York:
1169:(2nd ed.), Berlin, New York:
25:
1034:which is a simple noncommutative
761:is semisimple if and only if the
430:is semisimple if and only if any
64:. Some important rings, such as
27:Direct sum of irreducible modules
962:
566:{\displaystyle B\cong A\oplus C}
140:, the following are equivalent:
1194:(2nd ed.), W. H. Freeman,
1305:. New York. pp. 235â248.
616:
610:
601:
595:
483:
451:
207:. On the other hand, the ring
1:
1281:Graduate Texts in Mathematics
1214:Graduate Texts in Mathematics
870:is a positive integer, and M
414:A ring is said to be (left-)
371:can also be thought of as a
281:is a semisimple module over
258:if, for any field extension
223:Semisimple is stronger than
80:, are semisimple rings. An
40:, especially in the area of
1311:10.1007/978-1-4614-1231-1_8
779:group representation theory
750:is a finite group of order
242:be an algebra over a field
54:completely reducible module
1382:
1165:Bourbaki, Nicolas (2012),
1049:
915:
677:Semisimple rings are both
516:such that the composition
133:(irreducible) submodules.
29:
1226:10.1007/978-1-4419-8616-0
1081:artinian semisimple rings
888:matrices with entries in
777:, an important result in
534:is the identity. The map
233:indecomposable submodules
220:is not a direct summand.
102:Semisimple representation
786:WedderburnâArtin theorem
654:is semisimple, then all
645:von Neumann regular ring
172:, there is a complement
86:ArtinâWedderburn theorem
1208:Lam, Tsit-Yuen (2001),
576:or in more exact terms
225:completely decomposable
1025:
880:) denotes the ring of
626:
567:
493:
164:: for every submodule
1275:Pierce, R.S. (1982),
1026:
627:
568:
494:
256:absolutely semisimple
246:. Then a left module
1277:Associative Algebras
1083:to avoid confusion.
952:
583:
545:
445:
432:short exact sequence
363:A semisimple module
331:are also semisimple.
123:completely reducible
1150:, p. 133, VIII
1058:Jacobson semisimple
1046:Jacobson semisimple
478:
465:
434:of left (or right)
424:homological algebra
403:von Neumann regular
156:Every submodule of
78:characteristic zero
1105:Semisimple algebra
1052:Semiprimitive ring
1021:
701:semisimple algebra
622:
563:
489:
392:semiprimitive ring
358:Endomorphism rings
347:finitely generated
309:is semisimple and
32:Semisimple algebra
1294:978-1-4757-0165-4
1235:978-0-387-95325-0
1201:978-0-7167-1933-5
1180:978-3-540-35315-7
1056:A ring is called
1017:
775:Maschke's theorem
699:, it is called a
479:
466:
426:: namely, a ring
399:endomorphism ring
379:into the ring of
373:ring homomorphism
50:semisimple module
16:(Redirected from
1373:
1346:
1340:
1332:
1297:
1271:
1246:
1204:
1192:Basic algebra II
1188:Jacobson, Nathan
1183:
1151:
1145:
1139:
1133:
1127:
1121:
1072:Jacobson radical
1030:
1028:
1027:
1022:
1015:
987:
982:
965:
847:
788:, a unital ring
769:does not divide
714:commutative ring
687:Jacobson radical
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628:
623:
572:
570:
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410:Semisimple rings
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42:abstract algebra
21:
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1285:Springer-Verlag
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1259:Springer-Verlag
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1218:Springer-Verlag
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1171:Springer-Verlag
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1142:
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1113:
1096:
1054:
1048:
950:
949:
934:is semisimple.
920:
914:
898:
875:
869:
856:
845:
836:
835:
825:
818:
817:
809:
802:
801:
793:
746:is a field and
737:Krull dimension
733:Noetherian ring
718:semisimple ring
709:
581:
580:
543:
542:
517:
503:
443:
442:
412:
360:
322:
302:
291:
282:
276:
267:
214:
208:
177:
125:) if it is the
111:
90:direct products
62:semisimple ring
34:
28:
23:
22:
18:Semisimple ring
15:
12:
11:
5:
1379:
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1369:
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1268:978-0387953854
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1140:
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1115:
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1092:
1050:Main article:
1047:
1044:
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1008:
1005:
1002:
999:
996:
993:
990:
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978:
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969:
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960:
957:
941:, such as the
916:Main article:
913:
910:
909:
908:
896:
893:
871:
865:
852:
841:
831:
827:
823:
815:
811:
807:
799:
795:
782:
763:characteristic
740:
708:
705:
633:
632:
621:
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615:
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609:
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603:
600:
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559:
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550:
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499:
488:
485:
482:
477:
473:
469:
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460:
456:
453:
450:
411:
408:
407:
406:
395:
359:
356:
355:
354:
339:
332:
301:
298:
287:
272:
254:is said to be
192:
191:
162:direct summand
154:
148:
110:
107:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1378:
1367:
1364:
1362:
1361:Module theory
1359:
1358:
1356:
1344:
1338:
1330:
1326:
1322:
1320:9781461412311
1316:
1312:
1308:
1304:
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1270:
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1260:
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1252:
1248:
1245:
1241:
1237:
1231:
1227:
1223:
1219:
1215:
1211:
1206:
1203:
1197:
1193:
1189:
1185:
1182:
1176:
1172:
1168:
1167:Algèbre Ch. 8
1163:
1162:
1157:
1149:
1148:Bourbaki 2012
1144:
1141:
1138:, p. 125
1137:
1136:Sengupta 2012
1132:
1129:
1125:
1120:
1117:
1110:
1106:
1103:
1101:
1098:
1097:
1093:
1091:
1089:
1084:
1082:
1078:
1077:artinian ring
1073:
1069:
1068:
1067:semiprimitive
1063:
1059:
1053:
1045:
1043:
1041:
1040:module theory
1037:
1018:
1009:
1006:
1003:
1000:
997:
994:
991:
984:
976:
973:
970:
958:
955:
948:
947:
946:
944:
940:
939:Weyl algebras
935:
933:
929:
925:
919:
911:
906:
902:
894:
891:
887:
883:
879:
874:
868:
864:
860:
859:division ring
855:
851:
848:, where each
844:
840:
834:
830:
822:
814:
806:
798:
791:
787:
783:
780:
776:
772:
768:
764:
760:
757:
753:
749:
745:
741:
738:
734:
731:
727:
723:
719:
715:
711:
710:
706:
704:
702:
698:
695:
690:
688:
684:
680:
675:
673:
669:
665:
661:
657:
653:
648:
646:
642:
638:
619:
613:
607:
604:
598:
592:
589:
586:
579:
578:
577:
560:
557:
554:
551:
548:
541:
540:
539:
537:
532:
528:
524:
520:
514:
510:
506:
502:there exists
486:
480:
475:
471:
467:
462:
458:
454:
448:
441:
440:
439:
437:
433:
429:
425:
420:
417:
409:
404:
400:
396:
393:
389:
385:
384:endomorphisms
382:
381:abelian group
378:
374:
370:
366:
362:
361:
357:
352:
348:
344:
340:
337:
334:An arbitrary
333:
329:
325:
320:
316:
312:
308:
304:
303:
299:
297:
294:
290:
285:
279:
275:
270:
265:
261:
257:
253:
249:
245:
241:
236:
234:
230:
227:, which is a
226:
221:
218:
211:
206:
201:
199:
198:
188:
184:
180:
175:
171:
167:
163:
159:
155:
152:
149:
146:
143:
142:
141:
139:
136:For a module
134:
132:
128:
124:
120:
116:
108:
106:
104:
103:
97:
95:
91:
87:
83:
82:Artinian ring
79:
75:
71:
70:finite groups
67:
63:
59:
55:
51:
47:
46:module theory
43:
39:
33:
19:
1302:
1276:
1254:
1209:
1191:
1166:
1143:
1131:
1126:, p. 62
1119:
1087:
1085:
1080:
1065:
1062:J-semisimple
1061:
1057:
1055:
1033:
942:
936:
931:
927:
923:
921:
912:Simple rings
904:
900:
889:
885:
881:
877:
872:
866:
862:
853:
849:
842:
838:
832:
828:
820:
812:
804:
796:
789:
770:
766:
758:
751:
747:
743:
691:
676:
667:
663:
659:
655:
651:
649:
634:
575:
535:
530:
526:
522:
518:
512:
508:
504:
501:
435:
427:
421:
415:
413:
387:
376:
368:
367:over a ring
364:
342:
327:
323:
318:
310:
306:
292:
288:
283:
277:
273:
268:
263:
259:
255:
251:
247:
243:
239:
237:
222:
216:
209:
205:vector space
202:
195:
193:
186:
182:
178:
173:
169:
165:
157:
150:
144:
137:
135:
122:
118:
112:
100:
98:
94:matrix rings
61:
53:
49:
35:
1366:Ring theory
1251:Lang, Serge
918:Simple ring
826:) Ă ... Ă M
754:, then the
666:, that is,
66:group rings
38:mathematics
1355:Categories
1158:References
773:. This is
756:group ring
728:; being a
683:Noetherian
672:Kasch ring
670:is a left
641:projective
416:semisimple
336:direct sum
300:Properties
229:direct sum
176:such that
127:direct sum
119:semisimple
109:Definition
30:See also:
1337:cite book
1329:769756134
1111:Citations
1013:⟩
1007:−
998:−
989:⟨
980:⟩
968:⟨
945:-algebra
861:and each
689:is zero.
637:injective
605:⊕
590:≅
558:⊕
552:≅
484:→
452:→
341:A module
315:submodule
44:known as
1253:(2002),
1190:(1989),
1124:Lam 2001
1094:See also
722:artinian
720:; being
707:Examples
679:Artinian
525: :
507: :
472:→
459:→
353:is zero.
1255:Algebra
1244:1838439
784:By the
730:reduced
726:reduced
697:subring
694:central
351:radical
317:, then
1327:
1317:
1291:
1265:
1242:
1232:
1198:
1177:
1036:domain
1016:
712:For a
131:simple
115:module
74:fields
1100:Socle
857:is a
810:) Ă M
375:from
313:is a
250:over
160:is a
72:over
1343:link
1325:OCLC
1315:ISBN
1289:ISBN
1263:ISBN
1230:ISBN
1196:ISBN
1175:ISBN
1060:(or
884:-by-
724:and
681:and
639:and
397:The
321:and
238:Let
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58:ring
48:, a
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1222:doi
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765:of
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