Definition

Let R be a commutative ring with unity, and let M and N be two R-modules.

A pairing is any R-bilinear map . That is, it satisfies

e(rm,n) = e(m,rn) = re(m,n)

for any . Or equivalently, a pairing is an R-linear map

where denotes the tensor product of M and N.

A pairing can also be considered as an R-linear map , which matches the first definition by setting Φ(m)(n): = e(m,n).

A pairing is called perfect if the above map Φ is an isomorphism of R-modules.

Examples

Any scalar product on a real vector space V is a pairing (set M=N=V, R=R in the above definitions).

The determinant map (2 by 2 matrices over k)-->k can be seen as a pairing .

Slightly different usages of the notion of pairing

Scalar products on complex vector spaces are sometimes called pairings, although they are not bilinear. For example, in representation theory, one has a scalar product on the characters of complex representations of a finite group which is frequently called character pairing.