Bilinear map - Knowlpedia

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:

\beta(m+m',n)=\beta(m,n)+\beta(m',n),\quad \beta(m,n+n')=\beta(m,n)+\beta(m,n'),

\beta(rm,n)=r\,\beta(m,n),\quad \beta(m,rn)=r\,\beta(m,n).

Balanced maps for tensor products

Let $R$ be an arbitrary ring, let $M$ be a right $R$ -module, let $N$ be a left $R$ -module, and let $A$ be an abelian group. A map $\beta:M\times N\to A$ is $R$ -balanced if it is additive in each variable and

\beta(mr,n)=\beta(m,rn)

for every $m\in M$ , $n\in N$ , and $r\in R$ . No $R$ -module structure on $A$ is required.

These biadditive balanced maps are the maps represented by the universal property of the tensor product: they correspond uniquely to homomorphisms of abelian groups

M\otimes_RN\longrightarrow A.

Thus ordinary $R$ -bilinearity over a commutative ring and balancedness for a right-left module pair are related but distinct notions.

Examples

For any ring $R$ , multiplication $\mu:R\times R\to R$ is biadditive and $R$ -balanced when its codomain is viewed as an abelian group. If $R$ is commutative, multiplication is also $R$ -bilinear in the sense of the first definition.
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.