897:. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the
846:
977:
to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the
761:
507:
774:
is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for
796:
939:
ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over
989:(whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and
1267:
1215:
714:
970:
1144:
870:. One particularly useful condition is that the length of the sequence is finite and each quotient module
928:
990:
932:
790:
has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules
1167:
952:
913:
342:
1008:
1002:
1262:
1207:
1201:
921:
917:
851:
In order to prove the fact this way, one needs conditions on this sequence and on the modules
1112:
A consequence of the
Jacobson density theorem is Wedderburn's theorem; namely that any right
948:
905:
538:
519:
986:
902:
706:
488:
131:
36:
8:
613:
484:
476:
416:
142:
936:
767:
692:
531:
76:
29:
1229:
1221:
1211:
1162:
1156:
944:
655:
974:
935:
describe the relationships amongst all composition series of a single module. The
664:
311:
174:
982:
940:
252:
213:
1098:
56:
1256:
1225:
1132:
1113:
898:
659:
545:
358:
228:
158:
112:
91:
60:
1233:
963:
639:
128:
124:
84:
80:
1120:
1116:
993:
where the associated graph has a vertex for every indecomposable module.
978:
496:
44:
21:
17:
909:
671:
The converse of Schur's lemma is not true in general. For example, the
500:
412:
1140:
654:
is either the zero homomorphism or an isomorphism. Consequently, the
593:
48:
679:
is not simple, but its endomorphism ring is isomorphic to the field
551:
Not every module has a simple submodule; consider for instance the
244:
120:
107:
75:. Simple modules form building blocks for the modules of finite
1159:
are modules that can be written as a sum of simple submodules
1007:
An important advance in the theory of simple modules was the
522:
is to understand the irreducible representations of groups.
841:{\displaystyle \cdots \subset M_{2}\subset M_{1}\subset M.}
612:
is either the zero homomorphism or surjective because the
119:-module is an abelian group which has no non-zero proper
90:
In this article, all modules will be assumed to be right
686:
570:
be (left or right) modules over the same ring, and let
799:
717:
525:
181:
is not minimal, then there is a non-zero right ideal
1101:
may be viewed as (that is, isomorphic to) a ring of
955:
provides better arithmetic control, and uses simple
650:
is simple, then the prior two statements imply that
436:. The kernel of this homomorphism is a right ideal
889:is simple. In this case the sequence is called a
840:
755:
701:is a module which has a non-zero proper submodule
1254:
530:The simple modules are precisely the modules of
996:
534:1; this is a reformulation of the definition.
373:is a simple module. Conversely, suppose that
284:is not maximal, then there is a right ideal
541:, but the converse is in general not true.
153:is simple as a right module if and only if
548:, that is it is generated by one element.
381:-module. Then, for any non-zero element
756:{\displaystyle 0\to N\to M\to M/N\to 0.}
456:. By the above paragraph, we find that
365:. By the above paragraph, any quotient
1203:Linear Representations of Finite Groups
1011:. The Jacobson density theorem states:
962:modules to understand the structure of
943:, this is no loss as every module is a
268:is a right ideal which is not equal to
1255:
1206:. New York: Springer-Verlag. pp.
901:shows that the endomorphism ring of a
460:is a maximal right ideal. Therefore,
1199:
687:Simple modules and composition series
559:in light of the first example above.
444:, and a standard theorem states that
1139:. This can also be established as a
169:, then it is also a right ideal, so
592:is either the zero homomorphism or
13:
526:Basic properties of simple modules
235:is a non-zero proper submodule of
165:is a non-zero proper submodule of
14:
1279:
1057:-linearly independent subset of
518:representations. A major aim of
280:is not maximal. Conversely, if
79:, and they are analogous to the
1061:. Then there exists an element
464:is isomorphic to a quotient of
1240:
1193:
1180:
920:of finite length modules is a
747:
733:
727:
721:
508:main page on this relationship
1:
1246:Isaacs, Theorem 13.14, p. 185
1173:
971:Modular representation theory
584:be a module homomorphism. If
997:The Jacobson density theorem
272:and which properly contains
47:and have no non-zero proper
7:
1200:Serre, Jean-Pierre (1977).
1188:Non-commutative Ring Theory
1150:
933:Schreier refinement theorem
514:-modules are also known as
101:
10:
1284:
1168:Irreducible representation
1105:-linear operators on some
1000:
985:in various ways including
690:
662:. This result is known as
658:of any simple module is a
468:by a maximal right ideal.
51:. Equivalently, a module
953:Ordinary character theory
223:is simple if and only if
161:non-zero right ideal: If
111:-modules are the same as
1145:Artin–Wedderburn theorem
1119:is isomorphic to a full
1009:Jacobson density theorem
1003:Jacobson density theorem
193:is a right submodule of
35:are the (left or right)
991:Auslander–Reiten theory
544:Every simple module is
537:Every simple module is
389:, the cyclic submodule
922:Krull-Schmidt category
842:
757:
596:because the kernel of
506:(for details, see the
401:. The statement that
314:which is not equal to
185:properly contained in
1268:Representation theory
929:Jordan–Hölder theorem
914:Krull–Schmidt theorem
912:, so that the strong
906:indecomposable module
843:
766:A common approach to
758:
520:representation theory
411:is equivalent to the
1045:-linear operator on
951:of simple modules.
797:
715:
707:short exact sequence
489:group representation
292:. The quotient map
288:properly containing
208:is a right ideal of
1097:In particular, any
981:and describing the
1157:Semisimple modules
1019:be a simple right
937:Grothendieck group
891:composition series
838:
753:
705:, then there is a
693:Composition series
620:is a submodule of
600:is a submodule of
20:, specifically in
1163:Irreducible ideal
975:Brauer characters
945:semisimple module
656:endomorphism ring
448:is isomorphic to
173:is not minimal.
123:. These are the
1275:
1247:
1244:
1238:
1237:
1197:
1191:
1184:
1131:matrices over a
1023:-module and let
941:semisimple rings
887: + 1
868: + 1
847:
845:
844:
839:
828:
827:
815:
814:
762:
760:
759:
754:
743:
633:
608:is simple, then
588:is simple, then
583:
427:
410:
333:
324:, and therefore
323:
309:
267:
231:right ideal: If
66:
61:cyclic submodule
1283:
1282:
1278:
1277:
1276:
1274:
1273:
1272:
1253:
1252:
1251:
1250:
1245:
1241:
1218:
1198:
1194:
1185:
1181:
1176:
1153:
1032:
1005:
999:
983:module category
888:
878:
869:
859:
823:
819:
810:
806:
798:
795:
794:
739:
716:
713:
712:
695:
689:
625:
571:
528:
419:
402:
397:. Fix such an
361:right ideal of
334:is not simple.
325:
315:
310:has a non-zero
293:
255:
214:quotient module
201:is not simple.
104:
64:
63:generated by a
12:
11:
5:
1281:
1271:
1270:
1265:
1249:
1248:
1239:
1216:
1192:
1178:
1177:
1175:
1172:
1171:
1170:
1165:
1160:
1152:
1149:
1099:primitive ring
1095:
1094:
1028:
1001:Main article:
998:
995:
916:holds and the
883:
874:
864:
855:
849:
848:
837:
834:
831:
826:
822:
818:
813:
809:
805:
802:
764:
763:
752:
749:
746:
742:
738:
735:
732:
729:
726:
723:
720:
691:Main article:
688:
685:
539:indecomposable
527:
524:
510:). The simple
345:to a quotient
276:. Therefore,
115:, so a simple
113:abelian groups
103:
100:
92:unital modules
57:if and only if
26:simple modules
9:
6:
4:
3:
2:
1280:
1269:
1266:
1264:
1263:Module theory
1261:
1260:
1258:
1243:
1235:
1231:
1227:
1223:
1219:
1213:
1209:
1205:
1204:
1196:
1190:, Lemma 1.1.3
1189:
1183:
1179:
1169:
1166:
1164:
1161:
1158:
1155:
1154:
1148:
1146:
1142:
1138:
1134:
1133:division ring
1130:
1126:
1122:
1118:
1115:
1110:
1108:
1104:
1100:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1031:
1026:
1022:
1018:
1014:
1013:
1012:
1010:
1004:
994:
992:
988:
984:
980:
976:
972:
968:
965:
964:finite groups
961:
958:
954:
950:
946:
942:
938:
934:
930:
925:
923:
919:
915:
911:
907:
904:
903:finite length
900:
899:Fitting lemma
896:
892:
886:
882:
877:
873:
867:
863:
858:
854:
835:
832:
829:
824:
820:
816:
811:
807:
803:
800:
793:
792:
791:
789:
785:
781:
777:
773:
770:a fact about
769:
750:
744:
740:
736:
730:
724:
718:
711:
710:
709:
708:
704:
700:
694:
684:
682:
678:
674:
669:
667:
666:
665:Schur's lemma
661:
660:division ring
657:
653:
649:
645:
641:
637:
632:
628:
623:
619:
615:
611:
607:
603:
599:
595:
591:
587:
582:
578:
574:
569:
565:
560:
558:
554:
549:
547:
542:
540:
535:
533:
523:
521:
517:
513:
509:
505:
502:
498:
494:
490:
486:
482:
478:
474:
469:
467:
463:
459:
455:
451:
447:
443:
439:
435:
431:
426:
422:
418:
414:
409:
405:
400:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
337:Every simple
335:
332:
328:
322:
318:
313:
308:
304:
300:
296:
291:
287:
283:
279:
275:
271:
266:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
215:
211:
207:
202:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
160:
156:
152:
148:
144:
140:
135:
133:
130:
126:
125:cyclic groups
122:
118:
114:
110:
109:
99:
97:
93:
88:
86:
82:
81:simple groups
78:
74:
70:
62:
58:
54:
50:
46:
42:
38:
34:
31:
27:
23:
19:
1242:
1202:
1195:
1187:
1182:
1136:
1128:
1124:
1111:
1106:
1102:
1096:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1053:be a finite
1050:
1046:
1042:
1038:
1034:
1029:
1024:
1020:
1016:
1006:
966:
959:
956:
926:
894:
890:
884:
880:
875:
871:
865:
861:
856:
852:
850:
787:
783:
779:
775:
771:
765:
702:
698:
696:
680:
676:
672:
670:
663:
651:
647:
643:
640:endomorphism
635:
630:
626:
621:
617:
609:
605:
601:
597:
589:
585:
580:
576:
572:
567:
563:
561:
556:
552:
550:
543:
536:
529:
515:
511:
503:
492:
480:
472:
470:
465:
461:
457:
453:
449:
445:
441:
437:
433:
429:
424:
420:
417:homomorphism
413:surjectivity
407:
403:
398:
394:
390:
386:
382:
378:
377:is a simple
374:
370:
366:
362:
354:
350:
346:
338:
336:
330:
326:
320:
316:
306:
302:
298:
294:
289:
285:
281:
277:
273:
269:
264:
260:
256:
253:quotient map
248:
240:
236:
232:
224:
220:
216:
209:
205:
203:
198:
194:
190:
186:
182:
178:
170:
166:
162:
154:
150:
146:
138:
136:
116:
106:
105:
95:
94:over a ring
89:
85:group theory
72:
68:
52:
40:
32:
25:
15:
1121:matrix ring
1117:simple ring
979:Ext functor
516:irreducible
497:left module
428:that sends
393:must equal
341:-module is
243:, then the
212:, then the
141:is a right
67:element of
22:ring theory
18:mathematics
1257:Categories
1217:0387901906
1186:Herstein,
1174:References
1069:such that
949:direct sum
910:local ring
501:group ring
343:isomorphic
251:under the
175:Conversely
55:is simple
49:submodules
1226:0072-5285
1141:corollary
1135:for some
947:and so a
830:⊂
817:⊂
804:⊂
801:⋯
748:→
734:→
728:→
722:→
646:, and if
594:injective
499:over the
487:, then a
121:subgroups
43:that are
1151:See also
1114:Artinian
1109:-space.
1085:for all
1049:and let
931:and the
918:category
879: /
860: /
675:-module
575: :
555:-module
423:→
301:→
259:→
245:preimage
102:Examples
65:non-zero
45:non-zero
1234:2202385
1143:of the
1081:⋅
1073:⋅
1041:be any
1037:). Let
987:quivers
768:proving
634:, then
415:of the
359:maximal
229:maximal
159:minimal
149:, then
71:equals
37:modules
28:over a
1232:
1224:
1214:
786:. If
638:is an
546:cyclic
532:length
353:where
312:kernel
77:length
59:every
24:, the
1027:= End
973:uses
908:is a
624:. If
614:image
604:. If
495:is a
485:group
483:is a
477:field
475:is a
357:is a
227:is a
197:, so
177:, if
157:is a
143:ideal
132:order
129:prime
39:over
1230:OCLC
1222:ISSN
1212:ISBN
1127:-by-
1015:Let
927:The
893:for
778:and
566:and
562:Let
479:and
30:ring
1123:of
1089:in
1065:of
969:.
697:If
642:of
616:of
491:of
471:If
440:of
432:to
385:of
247:of
204:If
189:.
145:of
137:If
127:of
83:in
16:In
1259::
1228:.
1220:.
1210:.
1208:47
1147:.
1077:=
924:.
751:0.
683:.
668:.
629:=
579:→
434:xr
406:=
404:xR
391:xR
134:.
98:.
87:.
1236:.
1137:n
1129:n
1125:n
1107:D
1103:D
1093:.
1091:X
1087:x
1083:r
1079:x
1075:A
1071:x
1067:R
1063:r
1059:U
1055:D
1051:X
1047:U
1043:D
1039:A
1035:U
1033:(
1030:R
1025:D
1021:R
1017:U
967:G
960:G
957:C
895:M
885:i
881:M
876:i
872:M
866:i
862:M
857:i
853:M
836:.
833:M
825:1
821:M
812:2
808:M
788:N
784:N
782:/
780:M
776:N
772:M
745:N
741:/
737:M
731:M
725:N
719:0
703:N
699:M
681:Q
677:Q
673:Z
652:f
648:M
644:M
636:f
631:N
627:M
622:N
618:f
610:f
606:N
602:M
598:f
590:f
586:M
581:N
577:M
573:f
568:N
564:M
557:Z
553:Z
512:k
504:k
493:G
481:G
473:k
466:R
462:M
458:I
454:I
452:/
450:R
446:M
442:R
438:I
430:r
425:M
421:R
408:M
399:x
395:M
387:M
383:x
379:R
375:M
371:m
369:/
367:R
363:R
355:m
351:m
349:/
347:R
339:R
331:I
329:/
327:R
321:I
319:/
317:R
307:J
305:/
303:R
299:I
297:/
295:R
290:I
286:J
282:I
278:I
274:I
270:R
265:I
263:/
261:R
257:R
249:M
241:I
239:/
237:R
233:M
225:I
221:I
219:/
217:R
210:R
206:I
199:I
195:I
191:J
187:I
183:J
179:I
171:I
167:I
163:M
155:I
151:I
147:R
139:I
117:Z
108:Z
96:R
73:M
69:M
53:M
41:R
33:R
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