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In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring〔In this article, rings have a 1.〕 has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a 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 rings, the analogues for maximal left ideals and maximal right ideals also hold. * For pseudo-rings, the theorem holds for regular ideals. * 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Krull's theorem」の詳細全文を読む スポンサード リンク
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