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