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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 Regular local ring ： ウィキペディア英語版 Regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by Krull's principal ideal theorem ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''. The appellation ''regular'' is justified by the geometric meaning. A point ''x'' on an algebraic variety ''X'' is nonsingular if and only if the local ring $\backslash mathcal\_$ of germs at ''x'' is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings.〔A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.〕 For Noetherian local rings, there is the following chain of inclusions: ==Characterizations== There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if $A$ is a Noetherian local ring with maximal ideal $\backslash mathfrak$, then the following are equivalent definitions * Let $\backslash mathfrak\; =\; (a\_1,\; \backslash ldots,\; a\_n)$ where $n$ is chosen as small as possible. Then $A$ is regular if ::$\backslash mbox\; A\; =\; n\backslash ,$, :where the dimension is the Krull dimension. The minimal set of generators of $a\_1,\; \backslash ldots,\; a\_n$ are then called a ''regular system of parameters''. * Let $k\; =\; A\; /\; \backslash mathfrak$ be the residue field of $A$. Then $A$ is regular if ::$\backslash dim\_k\; \backslash mathfrak\; /\; \backslash mathfrak^2\; =\; \backslash dim\; A\backslash ,$, :where the second dimension is the Krull dimension. * Let $\backslash mbox\; A\; :=\; \backslash sup\; \backslash |\backslash mbox\; M\; \backslash mboxA\backslash mbox\; \backslash \}$ be the global dimension of $A$ (i.e., the supremum of the projective dimensions of all $A$-modules.) Then $A$ is regular if ::, :in which case, $\backslash mbox\; A\; =\; \backslash dim\; A$. 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「Regular local 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.