In mathematics, a pairing is an R-bilinear map from the Cartesian product of two R-modules, where the underlying ring R is commutative.

Let R be a commutative ring with unit, and let M, N and L be R-modules.

A pairing is any R-bilinear map ${\displaystyle e:M\times N\to L}$. That is, it satisfies

for any ${\displaystyle r\in R}$ and any ${\displaystyle m,m_{1},m_{2}\in M}$ and any ${\displaystyle n,n_{1},n_{2}\in N}$. Equivalently, a pairing is an R-linear map

where ${\displaystyle M\otimes _{R}N}$ denotes the tensor product of M and N.

A pairing can also be considered as an R-linear map ${\displaystyle \Phi :M\to \operatorname {Hom} _{R}(N,L)}$, which matches the first definition by setting ${\displaystyle \Phi (m)(n):=e(m,n)}$.

A pairing is called perfect if the above map ${\displaystyle \Phi }$ is an isomorphism of R-modules and the other evaluation map ${\displaystyle \Phi '\colon N\to \operatorname {Hom} _{R}(M,L)}$ is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field, M,N are finite dimensional vector spaces and L=R.

A pairing is called non-degenerate on the right if for the above map we have that ${\displaystyle e(m,n)=0}$ for all ${\displaystyle m}$ implies ${\displaystyle n=0}$; similarly, ${\displaystyle e}$ is called non-degenerate on the left if ${\displaystyle e(m,n)=0}$ for all ${\displaystyle n}$ implies ${\displaystyle m=0}$.