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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 Pairing ： ウィキペディア英語版 Pairing The concept of pairing treated here occurs in mathematics. ==Definition== Let ''R'' be a commutative ring with unity, and let ''M'', ''N'' and ''L'' be three ''R''-modules. A pairing is any ''R''-bilinear map $e:M\; \backslash times\; N\; \backslash to\; L$. That is, it satisfies :$e(rm,n)=e(m,rn)=re(m,n)$, :$e(m\_1+m\_2,n)=e(m\_1,n)+e(m\_2,n)$ and $e(m,n\_1+n\_2)=e(m,n\_1)+e(m,n\_2)$ for any $r\; \backslash in\; R$ and any $m,m\_1,m\_2\; \backslash in\; M$ and any $n,n\_1,n\_2\; \backslash in\; N$. Or equivalently, a pairing is an ''R''-linear map :$M\; \backslash otimes\_R\; N\; \backslash to\; L$ where $M\; \backslash otimes\_R\; N$ denotes the tensor product of ''M'' and ''N''. A pairing can also be considered as an R-linear map $\backslash Phi\; :\; M\; \backslash to\; \backslash operatorname\_\; (N,\; L)$, which matches the first definition by setting $\backslash Phi\; (m)\; (n)\; :=\; e(m,n)$. A pairing is called perfect if the above map $\backslash Phi$ is an isomorphism of R-modules. If $N=M$ a pairing is called alternating if for the above map we have $e(m,m)\; =\; 0$. A pairing is called non-degenerate if for the above map we have that $e(m,n)\; =\; 0$ for all $m$ implies $n=0$. 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「Pairing」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.