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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 Glossary of module theory ： ウィキペディア英語版 Glossary of module theory Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject. ==Basic definition== ; left R-module : A left module $M$ over the ring $R$ is an abelian group $(M,\; +)$ with an operation $R\; \backslash times\; M\; \backslash to\; M$ (called scalar multipliction) satisfies the following condition: ::$\backslash forall\; r,s\; \backslash in\; R,\; m,n\; \backslash in\; M$, :# $r\; (m\; +\; n)\; =\; rm\; +\; rn$ :# $r\; (s\; m)\; =\; (r\; s)\; m$ :# $1\_R\; \backslash ,\; m\; =\; m$ ; right R-module : A right module $M$ over the ring $R$ is an abelian group $(M,\; +)$ with an operation $M\; \backslash times\; R\; \backslash to\; M$ satisfies the following condition: :: $\backslash forall\; r,s\; \backslash in\; R,\; m,n\; \backslash in\; M$, :# $(m\; +\; n)\; r\; =\; m\; r\; +\; n\; r$ :# $(m\; s)\; r\; =\; r\; (s\; m)$ :# $m\; 1\_R\; =\; m$ : Or it can be defined as the left module $M$ over $R^\backslash textrm$ (the opposite ring of $R$). ; bimodule : If an abelian group $M$ is both a left $S$-module and right $R$-module, it can be made to a $(S,R)$-bimodule if $s(mr)\; =\; (sm)r\; \backslash ,\; \backslash forall\; s\; \backslash in\; S,\; r\; \backslash in\; R,\; m\; \backslash in\; M$. ; submodule : Given $M$ is a left $R$-module, a subgroup $N$ of $M$ is a submodule if $RN\; \backslash subseteq\; N$ . ; homomorphism of $R$-modules : For two left $R$-modules $M\_1,\; M\_2$, a group homomorphism $\backslash phi:\; M\_1\; \backslash to\; M\_2$ is called homomorphism of $R$-modules if $r\; \backslash phi(m)\; =\; \backslash phi\; (r\; m)\; \backslash ,\; \backslash forall\; r\; \backslash in\; R,\; m\; \backslash in\; M\_1$ . ; quotient module : Given a left $R$-modules $M$, a submodule $N$, $M/N$ can be made to a left $R$-module by $r(m+N)\; =\; rm\; +\; N\; \backslash ,\; \backslash forall\; r\; \backslash in\; R,\; m\; \backslash in\; M$ . It is also called a factor module. ; annihilator : The annihilator of a left $R$-module $M$ is the set $\backslash textrm(M)\; :=\; \backslash$ . It is a (left) ideal of $R$. : The annihilator of an element $m\; \backslash in\; M$ is the set $\backslash textrm(m)\; :=\; \backslash$. 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「Glossary of module theory」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.