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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 Gorenstein ring ： ウィキペディア英語版 Gorenstein ring In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring ''R'' with finite injective dimension, as an ''R''-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition. Gorenstein rings were introduced by Grothendieck, who named them because of their relation to a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings. Noncommutative analogues of 0-dimensional Gorenstein rings are called Frobenius rings. For Noetherian local rings, there is the following chain of inclusions. ==Definitions== A Gorenstein ring is a commutative ring such that each localization at a prime ideal is a Gorenstein local ring. The Gorenstein ring concept is a special case of the more general Cohen–Macaulay ring. The classical definition reads: A local Cohen–Macaulay ring ''R'' is called Gorenstein if there is a maximal ''R''-regular sequence in the maximal ideal generating an irreducible ideal. For a Noetherian commutative local ring $(R,\; m,\; k)$ of Krull dimension $n$, the following are equivalent: * $R$ has finite injective dimension as an $R$-module; * $R$ has injective dimension $n$ as an $R$-module; * $\backslash operatorname^i\_R\; (k,\; R)\; =\; 0$ for $i\; \backslash neq\; n$ and $\backslash operatorname^n\_R\; (k,\; R)$ is isomorphic to $k$; * $\backslash operatorname^i\_R\; (k,\; R)\; =\; 0$ for some $i\; >\; n$; * $\backslash operatorname^i\_R\; (k,\; R)\; =\; 0$ for all and $\backslash operatorname^n\_R\; (k,\; R)$ is isomorphic to $k$; * $R$ is an $n$-dimensional Gorenstein ring. A (not necessarily commutative) ring ''R'' is called Gorenstein if ''R'' has finite injective dimension both as a left ''R''-module and as a right ''R''-module. If ''R'' is a local ring, we say ''R'' is a local Gorenstein ring. 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「Gorenstein ring」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.