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2013-03-15T06:24:31Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q5290316

← Older revision		Revision as of 08:24, 15 March 2013
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	The ~~author~~'s ~~title~~ is ~~Andera~~ and ~~she thinks it seems fairly good~~. For a ~~whilst~~ I'~~ve been in Mississippi but now~~ I'~~m considering other options~~. ~~Office supervising~~ is ~~my profession~~. ~~What me~~ and ~~my family members adore~~ is ~~to climb but~~ I'~~m thinking on starting some thing new~~.<br><br>~~Here~~ is ~~my blog post~~ .~~.. clairvoyance;~~ [~~http:~~//si.~~dgmensa.org/xe/index.php?document_srl~~=~~48014&mid~~=~~c0102 si~~.~~dgmensa~~.~~org~~],		In [[mathematics]], and more specifically in [[ring theory]], '''Krull's theorem''', named after [[Wolfgang Krull]], asserts that a [[zero ring\|nonzero]] [[ring_(mathematics)\|ring]]<ref>In this article, rings have a 1.</ref> has at least one [[maximal ideal]]. The theorem was proved in 1929 by Krull, who used [[transfinite induction]].
			The theorem admits a [[Zorn's lemma#An example application\|simple proof using Zorn's lemma]], and in fact is equivalent to [[Zorn's lemma]],
			which in turn is equivalent to the [[axiom of choice]].

			==Variants==
			* For [[noncommutative ring]]s, the analogues for maximal left ideals and maximal right ideals also hold.
			* For [[pseudo-ring]]s, the theorem holds for [[regular ideal]]s.{{disambiguate\|date=December 2013}}
			* A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows:
			:::Let ''R'' be a ring, and let ''I'' be a [[proper ideal]] of ''R''. Then there is a maximal ideal of ''R'' containing ''I''.
			:This result implies the original theorem, by taking ''I'' to be the [[zero ideal]] (0). Conversely, applying the original theorem to ''R''/''I'' leads to this result.
			:To prove the stronger result directly, consider the set ''S'' of all proper ideals of ''R'' containing ''I''. The set ''S'' is nonempty since ''I'' ∈ ''S''. Furthermore, for any chain ''T'' of ''S'', the union of the ideals in ''T'' is an ideal ''J'', and a union of ideals not containing 1 does not contain 1, so ''J'' ∈ ''S''. By Zorn's lemma, ''S'' has a maximal element ''M''. This ''M'' is a maximal ideal containing ''I''.

			== Krull's Hauptidealsatz ==
			{{main\|Krull's principal ideal theorem}}
			Another theorem commonly referred to as Krull's theorem:
			:::Let <math>R</math> be a Noetherian ring and <math>a</math> an element of <math>R</math> which is neither a [[zero divisor]] nor a [[Unit (ring theory)\|unit]]. Then every minimal [[prime ideal]] <math>P</math> containing <math>a</math> has [[height (ring theory)\|height]] 1.

			==Notes==
			{{reflist}}

			== References ==
			* W. Krull, ''Idealtheorie in Ringen ohne Endlichkeitsbedingungen'', [[Mathematische Annalen]] '''10''' (1929), 729–744.

			[[Category:Ideals]]

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2010-07-18T00:29:46Z

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The author's title is Andera and she thinks it seems fairly good. For a whilst I've been in Mississippi but now I'm considering other options. Office supervising is my profession. What me and my family members adore is to climb but I'm thinking on starting some thing new.<br><br>Here is my blog post ... clairvoyance; [http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 si.dgmensa.org],

Dominating decision rule - Revision history

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q5290316

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