bilinear map (original) (raw)

Let R be a ring, and let M, N and P be modules over R. A function f:M×N→Pis said to be a _bilinear map_if for each b∈M the function h:N→Pdefined by h⁢(y)=f⁢(b,y) for all y∈N is linear (http://planetmath.org/LinearTransformation) (that is, an R-module homomorphism), and for each c∈N the function g:M→Pdefined by g⁢(x)=f⁢(x,c) for all x∈M is linear. Sometimes we may say that the function is R-bilinear, .

A common case is a bilinear map V×V→V, where V is a vector space over a field K; the vector space with this operation then forms an algebra over K.

If R is a commutative ring, then every R-bilinear map M×N→Pcorresponds in a natural way to a linear map M⊗N→P, where M⊗N is the tensor product of M and N (over R).