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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 Augmentation ideal ： ウィキペディア英語版 Augmentation ideal In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If ''G'' is a group and ''R'' a commutative ring, there is a ring homomorphism $\backslash varepsilon$, called the augmentation map, from the group ring : $R()$ to ''R'', defined by taking a sum : $\backslash sum\; r\_i\; g\_i$ to : $\backslash sum\; r\_i.$ Here ''r''''i'' is an element of ''R'' and ''g''''i'' an element of ''G''. The sums are finite, by definition of the group ring. In less formal terms, : $\backslash varepsilon(g)$ is defined as 1''R'' whatever the element ''g'' in ''G'', and $\backslash varepsilon$ is then extended to a homomorphism of ''R''-modules in the obvious way. The augmentation ideal is the kernel of $\backslash varepsilon$, and is therefore a two-sided ideal in ''R''(). It is generated by the differences : $g\; -\; g\text{'}$ of group elements. Furthermore it is also generated by : $g\; -\; 1\; ,\; g\backslash in\; G$ which is a basis for the augmentation ideal as a free ''R'' module. For ''R'' and ''G'' as above, the group ring ''R''() is an example of an ''augmented'' ''R''-algebra. Such an algebra comes equipped with a ring homomorphism to ''R''. The kernel of this homomorphism is the augmentation ideal of the algebra. Another class of examples of augmentation ideal can be the kernel of the counit $\backslash varepsilon$ of any Hopf algebra. The augmentation ideal plays a basic role in group cohomology, amongst other applications. ==References== * *Dummit and Foote, Abstract Algebra 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「Augmentation 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.