= Balanced map
For a <right module> $M_R$, a left <module> ${}_RN$, and an <abelian group> $P$, a balanced map is a map $b:M\times N\to P$ additive in each variable and satisfying
$$
b(mr,n)=b(m,rn).
$$
It is exactly the sort of map represented by the <tensor product of modules>. The balancing equation transfers a scalar between the two inputs without assuming that $R$ is commutative.
When $R$ is a <commutative ring> and $P$ is an $R$-<module>, an $R$-bilinear map is balanced and induces an $R$-linear map on the tensor product. Mere additive balancedness gives an additive induced map; it does not by itself require linearity in the target. For example, $(z,w)\mapsto\overline{zw}$ is additive and balanced over $\mathbb C$, but its induced map is conjugate-linear.
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