Pairing

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

,

and

for any and any and 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 .

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

If a pairing is called alternating if for the above map we have .

A pairing is called non-degenerate if for the above map we have that for all implies .