Pairing

In mathematics, a pairing is an R-bilinear map of modules, where R is the underlying ring.

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 ${\displaystyle e:M\times N\to L}$. That is, it satisfies

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

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