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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Serre's criterion for normality ： ウィキペディア英語版 Serre's criterion for normality In algebra, Serre's criterion for normality, introduced by Jean-Pierre Serre, gives necessary and sufficient conditions for a commutative Noetherian ring ''A'' to be a normal ring. The criterion involves the following two conditions for ''A'': *$R\_k:\; A\_$ of height ≤ ''k''. *$S\_k:\; \backslash operatorname\; A\_(\backslash mathfrak)\; \backslash \}$ for any prime ideal $\backslash mathfrak$. The statement is: *''A'' is a reduced ring $\backslash Leftrightarrow\; R\_0,\; S\_1$ hold. *''A'' is a normal ring $\backslash Leftrightarrow\; R\_1,\; S\_2$ hold. *''A'' is a Cohen–Macaulay ring $\backslash Leftrightarrow\; S\_k$ hold for all ''k''. Items 1, 3 trivially follow from the definitions. Item 2 is much deeper. For an integral domain, the criterion is due to Krull. The general case is due to Serre. == Proof == 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「Serre's criterion for normality」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.