414:. The Weyl algebra also gives an example of a simple algebra that is not a matrix algebra over a division algebra over its center: the Weyl algebra is infinite-dimensional, so Wedderburn's theorem does not apply.
395:
over fields. Simple algebras are building blocks of semisimple algebras: any finite-dimensional semisimple algebra is a
Cartesian product, in the sense of algebras, of finite-dimensional simple algebras.
425:: this says that every semisimple ring is a finite product of matrix rings over division rings. As a consequence of this generalization, every simple ring that is left- or right-
403:, and not every simple algebra is a semisimple algebra. However, every finite-dimensional simple algebra is a semisimple algebra, and every simple ring that is left- or right-
824:
802:
773:
751:
703:
677:
655:
633:
585:
501:
479:
457:
365:
729:
611:
284:
181:
145:
258:
878:
854:
560:
536:
362:
334:
314:
221:
201:
388:
776:
1078:
1024:
950:
960:
1057:
895:
924:
17:
916:
223:), but it has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns).
422:
400:
807:
785:
756:
734:
686:
660:
638:
616:
568:
484:
462:
440:
836:
The algebra of all linear transformations of an infinite-dimensional vector space over a field
708:
680:
590:
512:
263:
150:
114:
908:
237:
1096:
1034:
539:
66:
8:
516:
96:). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called
70:
59:
47:
863:
839:
545:
521:
392:
347:
319:
299:
206:
186:
43:
1074:
1053:
1020:
946:
920:
293:
1012:
988:
934:
890:
341:
93:
55:
28:
515:(sometimes called a Brauer algebra) is a simple finite-dimensional algebra over a
1066:
1049:
1030:
1008:
942:
857:
418:
373:
337:
111:
over a field does not have any nontrivial two-sided ideals (since any ideal of
1016:
230:, where every nonzero element has a multiplicative inverse, for instance, the
1090:
1000:
426:
404:
287:
227:
104:
89:
69:
of a simple ring is necessarily a field. It follows that a simple ring is an
830:
411:
369:
108:
32:
1041:
504:
377:
231:
51:
804:
is a central simple algebra, and is isomorphic to a matrix ring over
40:
992:
399:
One must be careful of the terminology: not every simple ring is a
979:
Henderson, D. W. (1965). "A short proof of
Wedderburn's theorem".
383:
Wedderburn proved these results in 1907 in his doctoral thesis,
391:. His thesis classified finite-dimensional simple and also
410:
An example of a simple ring that is not semisimple is the
88:) require in addition that a simple ring be left or right
372:
are rings of matrices over either the real numbers, the
829:
Every finite-dimensional central simple algebra over a
866:
842:
810:
788:
759:
737:
711:
689:
663:
641:
619:
593:
571:
548:
524:
487:
465:
443:
350:
322:
302:
266:
240:
209:
189:
153:
117:
316:
is a finite-dimensional simple algebra over a field
872:
848:
818:
796:
767:
745:
723:
697:
671:
649:
627:
605:
579:
554:
530:
495:
473:
451:
356:
328:
308:
278:
252:
215:
195:
175:
139:
364:. In particular, the only simple rings that are
1088:
961:"A short proof of the Wedderburn-Artin theorem"
833:is isomorphic to a matrix ring over that field.
389:Proceedings of the London Mathematical Society
103:Rings which are simple as rings but are not a
782:Every finite-dimensional simple algebra over
565:Every finite-dimensional simple algebra over
417:Wedderburn's result was later generalized to
226:An immediate example of a simple ring is a
978:
958:
915:. Colloquium publications. Vol. 24.
812:
790:
761:
739:
691:
665:
643:
621:
573:
489:
467:
445:
1065:
933:
85:
58:is a simple ring if and only if it is a
429:is a matrix ring over a division ring.
14:
1089:
1005:A First Course in Noncommutative Rings
907:
481:be the field of complex numbers, and
73:over this field. It is then called a
1040:
860:. It is also a simple algebra over
81:
999:
24:
1048:(3rd ed.), Berlin, New York:
1007:(2nd ed.), Berlin, New York:
941:(2nd ed.), Berlin, New York:
896:Simple algebra (universal algebra)
25:
1108:
880:that is not a semisimple algebra.
775:. These results follow from the
107:over themselves do exist: a full
1073:(2nd ed.), W. H. Freeman,
856:is a simple ring that is not a
705:is isomorphic to an algebra of
587:is isomorphic to an algebra of
959:Nicholson, William K. (1993).
459:be the field of real numbers,
170:
164:
134:
128:
13:
1:
917:American Mathematical Society
901:
54:and itself. In particular, a
819:{\displaystyle \mathbb {C} }
797:{\displaystyle \mathbb {C} }
768:{\displaystyle \mathbb {H} }
746:{\displaystyle \mathbb {R} }
698:{\displaystyle \mathbb {R} }
672:{\displaystyle \mathbb {H} }
650:{\displaystyle \mathbb {C} }
628:{\displaystyle \mathbb {R} }
580:{\displaystyle \mathbb {R} }
496:{\displaystyle \mathbb {H} }
474:{\displaystyle \mathbb {C} }
452:{\displaystyle \mathbb {R} }
7:
884:
432:
366:finite-dimensional algebras
286:matrices with entries in a
10:
1113:
80:Several references (e.g.,
1017:10.1007/978-1-4419-8616-0
724:{\displaystyle n\times n}
613:matrices with entries in
606:{\displaystyle n\times n}
279:{\displaystyle n\times n}
423:Wedderburn–Artin theorem
387:, which appeared in the
336:, it is isomorphic to a
176:{\displaystyle M_{n}(I)}
140:{\displaystyle M_{n}(R)}
385:On hypercomplex numbers
253:{\displaystyle n\geq 1}
874:
850:
820:
798:
769:
747:
731:matrices with entries
725:
699:
681:central simple algebra
673:
651:
629:
607:
581:
556:
532:
513:central simple algebra
497:
475:
453:
407:is a semisimple ring.
358:
330:
310:
296:proved that if a ring
280:
254:
217:
197:
177:
141:
46:that has no two-sided
913:Structure of Algebras
875:
851:
821:
799:
770:
748:
726:
700:
674:
652:
630:
608:
582:
557:
533:
498:
476:
454:
359:
331:
311:
281:
255:
218:
198:
178:
142:
864:
840:
808:
786:
757:
735:
709:
687:
661:
639:
617:
591:
569:
546:
522:
485:
463:
441:
348:
320:
300:
264:
238:
207:
187:
151:
115:
981:Amer. Math. Monthly
968:New Zealand J. Math
393:semisimple algebras
71:associative algebra
870:
846:
816:
794:
765:
743:
721:
695:
669:
647:
625:
603:
577:
552:
528:
493:
471:
449:
354:
326:
306:
276:
250:
213:
193:
173:
137:
1080:978-0-7167-1933-5
1026:978-0-387-95325-0
952:978-3-540-35315-7
935:Bourbaki, Nicolas
873:{\displaystyle k}
849:{\displaystyle k}
777:Frobenius theorem
555:{\displaystyle F}
531:{\displaystyle F}
357:{\displaystyle k}
329:{\displaystyle k}
309:{\displaystyle R}
294:Joseph Wedderburn
260:, the algebra of
216:{\displaystyle R}
196:{\displaystyle I}
92:(or equivalently
77:over this field.
16:(Redirected from
1104:
1083:
1071:Basic Algebra II
1067:Jacobson, Nathan
1062:
1037:
996:
975:
965:
955:
930:
891:Simple (algebra)
879:
877:
876:
871:
855:
853:
852:
847:
825:
823:
822:
817:
815:
803:
801:
800:
795:
793:
774:
772:
771:
766:
764:
752:
750:
749:
744:
742:
730:
728:
727:
722:
704:
702:
701:
696:
694:
678:
676:
675:
670:
668:
656:
654:
653:
648:
646:
634:
632:
631:
626:
624:
612:
610:
609:
604:
586:
584:
583:
578:
576:
561:
559:
558:
553:
537:
535:
534:
529:
502:
500:
499:
494:
492:
480:
478:
477:
472:
470:
458:
456:
455:
450:
448:
419:semisimple rings
363:
361:
360:
355:
342:division algebra
335:
333:
332:
327:
315:
313:
312:
307:
285:
283:
282:
277:
259:
257:
256:
251:
234:. Also, for any
222:
220:
219:
214:
202:
200:
199:
194:
182:
180:
179:
174:
163:
162:
146:
144:
143:
138:
127:
126:
56:commutative ring
29:abstract algebra
21:
1112:
1111:
1107:
1106:
1105:
1103:
1102:
1101:
1087:
1086:
1081:
1060:
1050:Springer-Verlag
1027:
1009:Springer-Verlag
993:10.2307/2313499
963:
953:
943:Springer-Verlag
927:
904:
887:
865:
862:
861:
858:semisimple ring
841:
838:
837:
811:
809:
806:
805:
789:
787:
784:
783:
760:
758:
755:
754:
738:
736:
733:
732:
710:
707:
706:
690:
688:
685:
684:
664:
662:
659:
658:
642:
640:
637:
636:
620:
618:
615:
614:
592:
589:
588:
572:
570:
567:
566:
547:
544:
543:
523:
520:
519:
488:
486:
483:
482:
466:
464:
461:
460:
444:
442:
439:
438:
435:
401:semisimple ring
374:complex numbers
349:
346:
345:
321:
318:
317:
301:
298:
297:
265:
262:
261:
239:
236:
235:
208:
205:
204:
188:
185:
184:
158:
154:
152:
149:
148:
147:is of the form
122:
118:
116:
113:
112:
86:Bourbaki (2012)
23:
22:
15:
12:
11:
5:
1110:
1100:
1099:
1085:
1084:
1079:
1063:
1059:978-0387953854
1058:
1038:
1025:
1001:Lam, Tsit-Yuen
997:
976:
956:
951:
931:
925:
919:. p. 37.
903:
900:
899:
898:
893:
886:
883:
882:
881:
869:
845:
834:
827:
814:
792:
780:
763:
741:
720:
717:
714:
693:
667:
645:
623:
602:
599:
596:
575:
563:
551:
527:
491:
469:
447:
434:
431:
353:
338:matrix algebra
325:
305:
275:
272:
269:
249:
246:
243:
212:
192:
172:
169:
166:
161:
157:
136:
133:
130:
125:
121:
75:simple algebra
31:, a branch of
18:Simple algebra
9:
6:
4:
3:
2:
1109:
1098:
1095:
1094:
1092:
1082:
1076:
1072:
1068:
1064:
1061:
1055:
1051:
1047:
1043:
1039:
1036:
1032:
1028:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
977:
973:
969:
962:
957:
954:
948:
944:
940:
939:Algèbre Ch. 8
936:
932:
928:
926:0-8218-1024-3
922:
918:
914:
910:
909:Albert, A. A.
906:
905:
897:
894:
892:
889:
888:
867:
859:
843:
835:
832:
828:
781:
778:
718:
715:
712:
682:
600:
597:
594:
564:
549:
541:
525:
518:
514:
510:
509:
508:
506:
430:
428:
424:
420:
415:
413:
408:
406:
402:
397:
394:
390:
386:
381:
379:
375:
371:
367:
351:
343:
339:
323:
303:
295:
291:
289:
288:division ring
273:
270:
267:
247:
244:
241:
233:
229:
228:division ring
224:
210:
190:
167:
159:
155:
131:
123:
119:
110:
106:
105:simple module
101:
99:
95:
91:
87:
83:
78:
76:
72:
68:
63:
61:
57:
53:
49:
45:
42:
38:
34:
30:
19:
1070:
1045:
1004:
984:
980:
971:
967:
938:
912:
831:finite field
436:
416:
412:Weyl algebra
409:
398:
384:
382:
370:real numbers
292:
225:
203:an ideal of
102:
98:quasi-simple
97:
79:
74:
64:
50:besides the
36:
26:
1097:Ring theory
1042:Lang, Serge
987:: 385–386.
505:quaternions
378:quaternions
290:is simple.
232:quaternions
109:matrix ring
94:semi-simple
82:Lang (2002)
37:simple ring
33:mathematics
902:References
340:over some
52:zero ideal
716:×
679:. Every
598:×
376:, or the
368:over the
271:×
245:≥
1091:Category
1069:(1989),
1044:(2002),
1003:(2001),
974:: 83–86.
937:(2012),
911:(2003).
885:See also
433:Examples
427:artinian
405:artinian
90:Artinian
41:non-zero
1046:Algebra
1035:1838439
421:in the
1077:
1056:
1033:
1023:
949:
923:
540:center
538:whose
67:center
964:(PDF)
683:over
657:, or
517:field
344:over
183:with
60:field
48:ideal
39:is a
1075:ISBN
1054:ISBN
1021:ISBN
947:ISBN
921:ISBN
503:the
437:Let
65:The
44:ring
35:, a
1013:doi
989:doi
753:or
542:is
84:or
27:In
1093::
1052:,
1031:MR
1029:,
1019:,
1011:,
985:72
983:.
972:22
970:.
966:.
945:,
635:,
511:A
507:.
380:.
100:.
62:.
1015::
995:.
991::
929:.
868:k
844:k
826:.
813:C
791:C
779:.
762:H
740:R
719:n
713:n
692:R
666:H
644:C
622:R
601:n
595:n
574:R
562:.
550:F
526:F
490:H
468:C
446:R
352:k
324:k
304:R
274:n
268:n
248:1
242:n
211:R
191:I
171:)
168:I
165:(
160:n
156:M
135:)
132:R
129:(
124:n
120:M
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.