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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 \times N \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 \in R and any m,m_1,m_2 \in M and any n,n_1,n_2 \in N . Or equivalently, a pairing is an ''R''-linear map
:M \otimes_R N \to L
where M \otimes_R N denotes the tensor product of ''M'' and ''N''.
A pairing can also be considered as an R-linear map
\Phi : M \to \operatorname_ (N, L) , which matches the first definition by setting
\Phi (m) (n) := e(m,n) .
A pairing is called perfect if the above map \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



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