182:
is nilpotent. In this case, the upper and lower nilradicals are equal, and moreover this ideal is the largest nilpotent ideal among nilpotent right ideals and among nilpotent left ideals.
137:
380:
359:
341:
528:
419:
465:
89:
85:
95:
257:
45:
500:
444:
492:
436:
376:
355:
337:
482:
474:
428:
262:
512:
456:
406:
508:
452:
402:
252:
41:
37:
414:
221:
188:: In view of the previous lemma, it is sufficient to show that the lower nilradical of
33:
522:
368:
390:
21:
17:
44:. Levitzky's theorem is one of the many results suggesting the veracity of the
487:
48:, and indeed provided a solution to one of Köthe's questions as described in (
496:
440:
232:. Since the lower nilradical contains all nilpotent ideals, it also contains
25:
504:
448:
162:
Every nonzero nil right ideal contains a nonzero nilpotent right ideal.
241:
165:
Every nonzero nil left ideal contains a nonzero nilpotent left ideal.
478:
432:
463:
Utumi, Yuzo (1963), "Mathematical Notes: A Theorem of
Levitzki",
151:
Any nil one-sided ideal is contained in the lower nil radical Nil
178:
be a right
Noetherian ring. Then every nil one-sided ideal of
354:(1st ed.), The Mathematical Association of America,
98:
52:). The result was originally submitted in 1939 as (
427:(3), The Johns Hopkins University Press: 437–442,
131:
56:), and a particularly simple proof was given in (
336:(1st ed.), Brooks/Cole Publishing Company,
473:(3), Mathematical Association of America: 286,
196:is right Noetherian, a maximal nilpotent ideal
417:(1945), "Solution of a problem of G. Koethe",
8:
126:
99:
68:This is Utumi's argument as it appears in (
40:, every nil one-sided ideal is necessarily
486:
97:
280:
53:
49:
273:
373:A First Course in Noncommutative Rings
292:
57:
7:
212:has no nonzero nilpotent ideals, so
316:
304:
132:{\displaystyle \{r\in R\mid ar=0\}}
69:
240:is equal to the lower nilradical.
14:
466:The American Mathematical Monthly
228:contains the lower nilradical of
420:American Journal of Mathematics
1:
295:, p. 210, Theorem 14.38
391:"On multiplicative systems"
283:, p. 37, Theorem 1.4.5
545:
334:Algebra, a graduate course
332:Isaacs, I. Martin (1993),
200:exists. By maximality of
86:ascending chain condition
36:, states that in a right
529:Theorems in ring theory
350:Herstein, I.N. (1968),
395:Compositio Mathematica
192:is nilpotent. Because
133:
389:Levitzki, J. (1950),
134:
352:Noncommutative rings
204:, the quotient ring
96:
20:, more specifically
375:, Springer-Verlag,
170:Levitzki's Theorem
488:10338.dmlcz/101274
129:
30:Levitzky's theorem
24:and the theory of
382:978-0-387-95183-6
536:
515:
490:
459:
410:
385:
364:
346:
320:
319:, Theorem 10.30.
314:
308:
302:
296:
290:
284:
278:
263:Jacobson radical
258:Köthe conjecture
138:
136:
135:
130:
46:Köthe conjecture
544:
543:
539:
538:
537:
535:
534:
533:
519:
518:
479:10.2307/2313127
462:
433:10.2307/2371958
415:Levitzki, Jakob
413:
388:
383:
367:
362:
349:
344:
331:
328:
323:
315:
311:
303:
299:
291:
287:
279:
275:
271:
253:Nilpotent ideal
249:
224:. As a result,
154:
94:
93:
66:
38:Noetherian ring
12:
11:
5:
542:
540:
532:
531:
521:
520:
517:
516:
460:
411:
386:
381:
365:
360:
347:
342:
327:
324:
322:
321:
309:
307:, Lemma 10.29.
297:
285:
272:
270:
267:
266:
265:
260:
255:
248:
245:
222:semiprime ring
172:
171:
167:
166:
163:
160:
152:
128:
125:
122:
119:
116:
113:
110:
107:
104:
101:
84:satisfies the
78:
77:
72:, p. 164-165)
65:
62:
34:Jacob Levitzki
32:, named after
13:
10:
9:
6:
4:
3:
2:
541:
530:
527:
526:
524:
514:
510:
506:
502:
498:
494:
489:
484:
480:
476:
472:
468:
467:
461:
458:
454:
450:
446:
442:
438:
434:
430:
426:
422:
421:
416:
412:
408:
404:
400:
396:
392:
387:
384:
378:
374:
370:
366:
363:
361:0-88385-015-X
357:
353:
348:
345:
343:0-534-19002-2
339:
335:
330:
329:
325:
318:
313:
310:
306:
301:
298:
294:
289:
286:
282:
281:Herstein 1968
277:
274:
268:
264:
261:
259:
256:
254:
251:
250:
246:
244:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
181:
177:
169:
168:
164:
161:
158:
150:
149:
148:
146:
142:
123:
120:
117:
114:
111:
108:
105:
102:
91:
87:
83:
75:
74:
73:
71:
63:
61:
59:
55:
54:Levitzki 1950
51:
50:Levitzki 1945
47:
43:
39:
35:
31:
27:
23:
19:
470:
464:
424:
418:
398:
394:
372:
351:
333:
312:
300:
288:
276:
237:
233:
229:
225:
217:
213:
209:
205:
201:
197:
193:
189:
185:
184:
179:
175:
173:
156:
144:
140:
92:of the form
90:annihilators
81:
80:Assume that
79:
67:
29:
15:
293:Isaacs 1993
22:ring theory
18:mathematics
326:References
58:Utumi 1963
26:nil ideals
497:0002-9890
441:0002-9327
401:: 76–80,
369:Lam, T.Y.
236:, and so
112:∣
106:∈
42:nilpotent
523:Category
371:(2001),
317:Lam 2001
305:Lam 2001
247:See also
70:Lam 2001
513:1532056
505:2313127
457:0012269
449:2371958
407:0033799
147:. Then
511:
503:
495:
455:
447:
439:
405:
379:
358:
340:
242:Q.E.D.
143:is in
139:where
501:JSTOR
445:JSTOR
269:Notes
220:is a
186:Proof
76:Lemma
64:Proof
493:ISSN
437:ISSN
377:ISBN
356:ISBN
338:ISBN
174:Let
483:hdl
475:doi
429:doi
88:on
60:).
16:In
525::
509:MR
507:,
499:,
491:,
481:,
471:70
469:,
453:MR
451:,
443:,
435:,
425:67
423:,
403:MR
397:,
393:,
159:);
28:,
485::
477::
431::
409:.
399:8
238:N
234:N
230:R
226:N
218:N
216:/
214:R
210:N
208:/
206:R
202:N
198:N
194:R
190:R
180:R
176:R
157:R
155:(
153:*
145:R
141:a
127:}
124:0
121:=
118:r
115:a
109:R
103:r
100:{
82:R
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