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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 Integral closure of an ideal ： ウィキペディア英語版 Integral closure of an ideal In algebra, the integral closure of an ideal ''I'' of a commutative ring ''R'', denoted by $\backslash overline$, is the set of all elements ''r'' in ''R'' that are integral over ''I'': there exist $a\_i\; \backslash in\; I^i$ such that :$r^n\; +\; a\_1\; r^\; +\; \backslash cdots\; +\; a\_\; r\; +\; a\_n\; =\; 0.$ It is similar to the integral closure of a subring. For example, if ''R'' is a domain, an element ''r'' in ''R'' belongs to $\backslash overline$ if and only if there is a finitely generated ''R''-module ''M'', annihilated only by zero, such that $r\; M\; \backslash subset\; I\; M$. It follows that $\backslash overline$ is an ideal of ''R'' (in fact, the integral closure of an ideal is always an ideal; see below.) ''I'' is said to be integrally closed if $I\; =\; \backslash overline$. The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring. == Examples == *In $\backslash mathbb(y)$, $x^i\; y^$ is integral over $(x^d,\; y^d)$. *Radical ideals (e.g., prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed. *Let $R\; =\; k(\backslash ldots,\; X\_n)$ be a polynomial ring over a field ''k''. An ideal ''I'' in ''R'' is called monomial if it is generated by monomials; i.e., $X\_1^\; \backslash cdots\; X\_n^$. The integral closure of a monomial ideal is monomial. 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「Integral closure of an ideal」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.