170:
In the case of commutative rings, there is always a maximal nil ideal: the nilradical of the ring. The existence of such a maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is again nil. However, the truth of the assertion that the sum of two
127:) = 0. Therefore, the set of all nilpotent elements forms an ideal known as the nil radical of a ring. Because the nil radical contains every nilpotent element, an ideal of a commutative ring is nil
51:
is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nilpotent elements does not always form an ideal for
187:, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil. There are two main barriers for nil ideals to be nilpotent:
221:
of the ring, and since the
Jacobson radical is a nilpotent ideal (due to the artinian hypothesis), the result follows. In fact, this has been generalized to right
71:
In commutative rings, the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving
143:
is nil. For a non commutative ring however, it is not in general true that the set of nilpotent elements forms an ideal, or that
131:
it is a subset of the nil radical, and so the nil radical is maximal among nil ideals. Furthermore, for any nilpotent element
399:
191:
There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponents may be required.
163:
The theory of nil ideals is of major importance in noncommutative ring theory. In particular, through the understanding of
367:
349:
431:
391:
167:—rings whose every element is nilpotent—one may obtain a much better understanding of more general rings.
380:
171:
left nil ideals is again a left nil ideal remains elusive; it is an open problem known as the
226:
409:
252:
44:
8:
52:
25:
242:
206:
Clearly both of these barriers must be avoided for a nil ideal to qualify as nilpotent.
172:
60:
376:
214:
29:
175:. The Köthe conjecture was first posed in 1930 and yet remains unresolved as of 2023.
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37:
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48:
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21:
17:
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and sums of nilpotent elements are both nilpotent. This is because if
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The notion of a nil ideal has a deep connection with that of a
229:. A particularly simple proof due to Utumi can be found in (
344:(1st ed.), The Mathematical Association of America,
198:
nilpotent elements may be nonzero for arbitrarily high
217:
by observing that any nil ideal is contained in the
55:. Nil ideals are still associated with interesting
388:International Congress of Mathematicians, Vol. II
423:
362:(1st ed.), Brooks/Cole Publishing Company,
178:
381:"Some results in noncommutative ring theory"
375:
300:
339:
312:
287:
230:
158:
424:
357:
324:
275:
213:, any nil ideal is nilpotent. This is
269:
151:is a nil (one-sided) ideal, even if
66:
119:= 0, and by the binomial theorem, (
13:
14:
443:
306:
293:
281:
1:
392:European Mathematical Society
333:
24:, a left, right or two-sided
179:Relation to nilpotent ideals
7:
236:
36:if each of its elements is
10:
448:
360:Algebra, a graduate course
358:Isaacs, I. Martin (1993),
83:are nilpotent elements of
59:, especially the unsolved
327:, Corollary 14.3, p. 195.
233:, Theorem 1.4.5, p. 37).
225:; the result is known as
340:Herstein, I. N. (1968),
263:
290:, Definition (b), p. 13
135:of a commutative ring
432:Ideals (ring theory)
394:, pp. 259–269,
342:Noncommutative rings
159:Noncommutative rings
53:noncommutative rings
20:, more specifically
377:Smoktunowicz, Agata
227:Levitzky's theorem
99:is any element of
73:nilpotent elements
401:978-3-03719-022-7
301:Smoktunowicz 2006
67:Commutative rings
439:
418:
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258:Jacobson radical
243:Köthe conjecture
223:noetherian rings
219:Jacobson radical
173:Köthe conjecture
61:Köthe conjecture
49:commutative ring
32:is said to be a
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248:Nilpotent ideal
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194:The product of
185:nilpotent ideal
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12:
11:
5:
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155:is nilpotent.
129:if and only if
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57:open questions
9:
6:
4:
3:
2:
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433:
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369:0-534-19002-2
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351:0-88385-015-X
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321:
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313:Herstein 1968
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299:Section 2 of
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288:Herstein 1968
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231:Herstein 1968
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211:artinian ring
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78:
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50:
46:
41:
39:
35:
31:
27:
23:
19:
413:, retrieved
387:
359:
341:
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308:
295:
283:
271:
208:
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199:
195:
182:
169:
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152:
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140:
139:, the ideal
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96:
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84:
80:
76:
70:
42:
33:
15:
325:Isaacs 1993
276:Isaacs 1993
209:In a right
22:ring theory
18:mathematics
415:2009-08-19
390:, Zürich:
334:References
253:Nilradical
45:nilradical
165:nil rings
95:= 0, and
38:nilpotent
34:nil ideal
426:Category
379:(2006),
303:, p. 260
278:, p. 194
237:See also
147: ·
103:, then (
91:= 0 and
410:2275597
315:, p. 21
408:
398:
366:
348:
215:proved
384:(PDF)
264:Notes
87:with
47:of a
28:of a
26:ideal
396:ISBN
364:ISBN
346:ISBN
111:) =
79:and
43:The
30:ring
40:.
16:In
428::
406:MR
404:,
386:,
141:aR
63:.
202:.
200:n
196:n
153:a
149:R
145:a
137:R
133:a
125:b
123:+
121:a
117:r
115:·
113:a
109:r
107:·
105:a
101:R
97:r
93:b
89:a
85:R
81:b
77:a
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