bilinear map

Let $R$ be a ring, and let $M$ , $N$ and $P$ be modules over $R$ . A function $f\colon M\times N\to P$ is said to be a bilinear map if for each $b\in M$ the function $h\colon N\to P$ defined by $h(y)=f(b,y)$ for all $y\in N$ is linear (that is, an $R$ -module homomorphism), and for each $c\in N$ the function $g\colon M\to P$ defined by $g(x)=f(x,c)$ for all $x\in M$ is linear. Sometimes we may say that the function is $R$ -bilinear, in order to make the base ring clear.

A common case is a bilinear map $V\times V\to V$ , 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 $M\times N\to P$ corresponds in a natural way to a linear map $M\otimes N\to P$ , where $M\otimes N$ is the tensor product of $M$ and $N$ (over $R$ ).