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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 ideal quotient ： ウィキペディア英語版 ideal quotient In abstract algebra, if ''I'' and ''J'' are ideals of a commutative ring ''R'', their ideal quotient (''I'' : ''J'') is the set :$(I\; :\; J)\; =\; \backslash$ Then (''I'' : ''J'') is itself an ideal in ''R''. The ideal quotient is viewed as a quotient because $IJ\; \backslash subset\; K$ if and only if $I\; \backslash subset\; K\; :\; J$. The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below). (''I'' : ''J'') is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal. ==Properties== The ideal quotient satisfies the following properties: *$(I\; :J)=\backslash mathrm\_R((J+I)/I)$ as $R$-modules, where $\backslash mathrm\_R(M)$ denotes the annihilator of $M$ as an $R$-module. *$J\; \backslash subset\; I\; \backslash Rightarrow\; I\; :\; J\; =\; R$ *$I\; :\; R\; =\; I$ *$R\; :\; I\; =\; R$ *$I\; :\; (J\; +\; K)\; =\; (I\; :\; J)\; \backslash cap\; (I\; :\; K)$ *$I\; :\; (r)\; =\; \backslash frac(I\; \backslash cap\; (r))$ (as long as ''R'' is an integral domain) 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「ideal quotient」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.