Bilinear map

A bilinear map between $R$-modules (for a commutative ring $R$) is a function $\beta\colon M\times N\to P$ from a cartesian product of $R$-modules such that, for each fixed argument, the resulting map is $R$-linear in the other:

For a noncommutative ring $R$, if $M$ is a right $R$-module and $N$ is a left $R$-module, one usually requires $\beta$ to be $R$-balanced, meaning additionally $\beta(mr,n)=\beta(m,rn)$.

Balanced bilinear maps are exactly the maps represented by the universal property of the tensor product : they correspond uniquely to homomorphisms out of $M\otimes_R N$ in the category of modules /abelian groups.

Examples:

• Multiplication $\mu\colon R\times R\to R$ is bilinear for any ring $R$.
• For a commutative ring $R$ and an $R$-module $M$, the evaluation pairing $M^\vee\times M\to R$ is bilinear, where $M^\vee$ is the dual module .