In mathematics, a pairing is an R-bilinear map from the Cartesian product of two R-modules, where the underlying ring R is commutative.
Definition Let R be a commutative ring with unit, and let M, N and L be R-modules. A pairing is any R-bilinear map e : M × N → L {\displaystyle e:M\times N\to L} . That is, it satisfies
e ( r ⋅ m , n ) = e ( m , r ⋅ n ) = r ⋅ e ( m , n ) {\displaystyle e(r\cdot m,n)=e(m,r\cdot n)=r\cdot e(m,n)} ,
e ( m 1 + m 2 , n ) = e ( m 1 , n ) + e ( m 2 , n ) {\displaystyle 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 ) {\displaystyle e(m,n_{1}+n_{2})=e(m,n_{1})+e(m,n_{2})}
for any r ∈ R {\displaystyle r\in R} and any m , m 1 , m 2 ∈ M {\displaystyle m,m_{1},m_{2}\in M} and any n , n 1 , n 2 ∈ N {\displaystyle n,n_{1},n_{2}\in N} . Equivalently, a pairing is an R-linear map
M ⊗ R N → L {\displaystyle M\otimes _{R}N\to L}
where M ⊗ R N {\displaystyle M\otimes _{R}N} denotes the tensor product of M and N. A pairing can also be considered as an R-linear map
Φ : M → Hom R ( N , L ) {\displaystyle \Phi :M\to \operatorname {Hom} _{R}(N,L)} , which matches the first definition by setting
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