293:
562:
535:
504:
414:
387:
324:
473:
445:
352:
166:. The analogous statement for left Artinian rings holds as well. This is not true in general for Artinian modules, because there are
720:
679:
745:
29:
359:
33:
585:
667:
231:
632:
577:
37:
611:
144:
117:
65:
41:
716:
675:
606:
417:
109:
221:
210:
148:
89:
17:
540:
513:
482:
392:
365:
302:
120:" are equivalent. Without the semiprimary condition, the only true implication is that if
601:
167:
163:
113:
77:
450:
422:
329:
725:
132:
739:
155:
140:
131:
The theorem takes its current form from a paper by
Charles Hopkins and a paper by
21:
708:
159:
584:
is a
Grothendieck category with an Artinian generator, then every Artinian
537:
has a finite composition series. Stacking the composition series from the
576:
Several generalizations and extensions of the theorem exist. One concerns
93:
730:
On rings which satisfy the minimum condition for the right-hand ideals
135:, both in 1939. For this reason it is often cited as the
668:"A Seventy Years Jubilee: The Hopkins-Levitzki Theorem"
181:
is left
Artinian if and only if it is left Noetherian.
162:
of the theorem is: a right
Artinian ring is also right
543:
516:
485:
453:
425:
395:
368:
332:
305:
234:
168:
examples of
Artinian modules which are not Noetherian
556:
529:
498:
467:
439:
408:
381:
346:
318:
287:
732:, Compositio Mathematica, v. 7, pp. 214–222.
564:end to end, we obtain a composition series for
8:
703:Rings with minimal condition for left ideals
548:
542:
521:
515:
490:
484:
475:is a semisimple ring. Furthermore, since
457:
452:
429:
424:
400:
394:
373:
367:
336:
331:
310:
304:
276:
267:
252:
239:
233:
479:is nilpotent, only finitely many of the
189:Here is the proof of the following: Let
696:Basic Algebra: Groups, Rings and Fields
623:
205:is either Artinian or Noetherian, then
713:A first course in noncommutative rings
705:, Ann. of Math. (2) 40, pages 712–730.
633:"Teilerkettensatz und Vielfachensatz"
288:{\displaystyle F_{i}=J^{i-1}M/J^{i}M}
108:-module, the three module conditions
7:
653:
173:Another direct corollary is that if
510:is Artinian (or Noetherian), then
14:
128:is both Noetherian and Artinian.
213:of this is true over any ring.)
26:Akizuki–Hopkins–Levitzki theorem
209:has a composition series. (The
143:is sometimes included since he
124:has a composition series, then
151:a few years earlier, in 1935.
1:
154:Since it is known that right
715:, Springer-Verlag. page 55
762:
637:Proc. Phys.-Math. Soc. Jpn
572:In Grothendieck categories
193:be a semiprimary ring and
158:are semiprimary, a direct
100:is a semiprimary ring and
30:descending chain condition
326:may then be viewed as an
34:ascending chain condition
177:is right Artinian, then
137:Hopkins–Levitzki theorem
48:(with 1) is called
746:Theorems in ring theory
701:Charles Hopkins (1939)
631:Akizuki, Yasuo (1935).
578:Grothendieck categories
672:Ring and Module Theory
670:. In Toma Albu (ed.).
558:
531:
500:
469:
441:
410:
383:
348:
320:
289:
559:
557:{\displaystyle F_{i}}
532:
530:{\displaystyle F_{i}}
501:
499:{\displaystyle F_{i}}
470:
442:
411:
409:{\displaystyle F_{i}}
384:
382:{\displaystyle F_{i}}
349:
321:
319:{\displaystyle F_{i}}
290:
541:
514:
483:
451:
423:
393:
366:
358:is contained in the
330:
303:
232:
694:Cohn, P.M. (2003),
468:{\displaystyle R/J}
440:{\displaystyle R/J}
347:{\displaystyle R/J}
666:Toma Albu (2010).
612:Composition series
554:
527:
496:
465:
437:
406:
379:
344:
316:
285:
118:composition series
607:Noetherian module
447:-module, because
149:commutative rings
40:over semiprimary
753:
698:
686:
685:
663:
657:
651:
645:
644:
628:
563:
561:
560:
555:
553:
552:
536:
534:
533:
528:
526:
525:
506:are nonzero. If
505:
503:
502:
497:
495:
494:
474:
472:
471:
466:
461:
446:
444:
443:
438:
433:
415:
413:
412:
407:
405:
404:
388:
386:
385:
380:
378:
377:
354:-module because
353:
351:
350:
345:
340:
325:
323:
322:
317:
315:
314:
294:
292:
291:
286:
281:
280:
271:
263:
262:
244:
243:
90:Jacobson radical
20:, in particular
18:abstract algebra
761:
760:
756:
755:
754:
752:
751:
750:
736:
735:
693:
690:
689:
682:
665:
664:
660:
656:, Theorem 5.3.9
652:
648:
630:
629:
625:
620:
602:Artinian module
598:
592:is Noetherian.
574:
544:
539:
538:
517:
512:
511:
486:
481:
480:
449:
448:
421:
420:
396:
391:
390:
369:
364:
363:
328:
327:
306:
301:
300:
272:
248:
235:
230:
229:
187:
185:Sketch of proof
147:the result for
96:states that if
78:nilpotent ideal
12:
11:
5:
759:
757:
749:
748:
738:
737:
734:
733:
726:Jakob Levitzki
723:
706:
699:
688:
687:
680:
658:
646:
622:
621:
619:
616:
615:
614:
609:
604:
597:
594:
573:
570:
551:
547:
524:
520:
493:
489:
464:
460:
456:
436:
432:
428:
403:
399:
376:
372:
343:
339:
335:
313:
309:
284:
279:
275:
270:
266:
261:
258:
255:
251:
247:
242:
238:
186:
183:
156:Artinian rings
133:Jacob Levitzki
88:) denotes the
13:
10:
9:
6:
4:
3:
2:
758:
747:
744:
743:
741:
731:
727:
724:
722:
721:0-387-95183-0
718:
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681:9783034600071
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397:
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357:
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192:
184:
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157:
152:
150:
146:
142:
141:Yasuo Akizuki
138:
134:
129:
127:
123:
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
31:
28:connects the
27:
23:
19:
729:
712:
702:
695:
674:. Springer.
671:
661:
649:
640:
636:
626:
589:
581:
575:
565:
507:
476:
355:
296:
225:
217:
215:
206:
202:
201:-module. If
198:
194:
190:
188:
178:
174:
172:
153:
136:
130:
125:
121:
105:
101:
97:
85:
81:
73:
69:
61:
57:
53:
49:
45:
25:
15:
360:annihilator
139:. However
116:and "has a
50:semiprimary
22:ring theory
643:: 337–345.
618:References
418:semisimple
164:Noetherian
110:Noetherian
66:semisimple
44:. A ring
709:T. Y. Lam
654:Cohn 2003
257:−
160:corollary
740:Category
596:See also
299:-module
211:converse
114:Artinian
80:, where
728:(1939)
711:(2001)
389:. Each
222:radical
220:be the
197:a left
94:theorem
92:. The
76:) is a
38:modules
719:
678:
586:object
295:. The
228:. Set
145:proved
104:is an
24:, the
580:: if
416:is a
170:.
64:) is
42:rings
717:ISBN
676:ISBN
216:Let
68:and
32:and
588:in
362:of
224:of
52:if
36:in
16:In
742::
641:17
639:.
635:.
568:.
112:,
684:.
590:G
582:G
566:M
550:i
546:F
523:i
519:F
508:M
492:i
488:F
477:J
463:J
459:/
455:R
435:J
431:/
427:R
402:i
398:F
375:i
371:F
356:J
342:J
338:/
334:R
312:i
308:F
297:R
283:M
278:i
274:J
269:/
265:M
260:1
254:i
250:J
246:=
241:i
237:F
226:R
218:J
207:M
203:M
199:R
195:M
191:R
179:R
175:R
126:M
122:M
106:R
102:M
98:R
86:R
84:(
82:J
74:R
72:(
70:J
62:R
60:(
58:J
56:/
54:R
46:R
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