309:
249:
17:
335:
278:
51:
97:
slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows:
76:
295:
43:
314:
287:
254:
63:
59:
55:
36:
329:
299:
87:
47:
115:
83:
28:
24:
318:
258:
142:
40:
79:, the analogues for maximal left ideals and maximal right ideals also hold.
276:
291:
196:, and a union of ideals not containing 1 does not contain 1, so
137:
The statement of the original theorem can be obtained by taking
156:
To prove the "stronger" result directly, consider the set
20:, a theorem on the height of ideals in a Noetherian Ring.
50:. The theorem was proved in 1929 by Krull, who used
145:(0). Conversely, applying the original theorem to
8:
310:Journal of the London Mathematical Society
250:Journal of the London Mathematical Society
307:Hodges, W. (1979). "Krull implies Zorn".
247:Hodges, W. (1979). "Krull implies Zorn".
230:
62:, which in turn is equivalent to the
7:
122:. Then there is a maximal ideal of
14:
237:In this article, rings have a 1.
58:, and in fact is equivalent to
56:simple proof using Zorn's lemma
18:Krull's principal ideal theorem
216:is a maximal ideal containing
180:. Furthermore, for any chain
1:
188:, the union of the ideals in
27:, and more specifically in
352:
15:
319:10.1112/jlms/s2-19.2.285
259:10.1112/jlms/s2-19.2.285
160:of all proper ideals of
86:, the theorem holds for
54:. The theorem admits a
16:Not to be confused with
313:. s2-19 (2): 285–287.
253:. s2-19 (2): 285–287.
208:has a maximal element
279:Mathematische Annalen
153:leads to this result.
52:transfinite induction
336:Ideals (ring theory)
204:. By Zorn's lemma,
77:noncommutative rings
110:be a ring, and let
292:10.1007/BF01454872
172:is nonempty since
46:has at least one
39:, asserts that a
343:
322:
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275:
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72:
64:axiom of choice
33:Krull's theorem
21:
12:
11:
5:
349:
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286:(1): 729–744.
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154:
135:
134:
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99:
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88:regular ideals
80:
71:
68:
37:Wolfgang Krull
35:, named after
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2:
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129:
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109:
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96:
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49:
48:maximal ideal
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34:
30:
26:
19:
308:
283:
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192:is an ideal
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177:
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157:
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146:
138:
127:
123:
119:
116:proper ideal
111:
107:
94:
84:pseudo-rings
60:Zorn's lemma
32:
22:
168:. The set
164:containing
126:containing
29:ring theory
25:mathematics
270:References
143:zero ideal
141:to be the
95:apparently
300:119883473
330:Category
212:. This
70:Variants
41:nonzero
298:
296:S2CID
225:Notes
114:be a
106:Let
82:For
75:For
44:ring
315:doi
288:doi
284:101
255:doi
184:of
118:of
93:An
23:In
332::
294:.
282:.
200:∈
176:∈
66:.
31:,
321:.
317::
302:.
290::
261:.
257::
220:.
218:I
214:M
210:M
206:S
202:S
198:J
194:J
190:T
186:S
182:T
178:S
174:I
170:S
166:I
162:R
158:S
151:I
149:/
147:R
139:I
130:.
128:I
124:R
120:R
112:I
108:R
90:.
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