 Universal property of the tensor product of modules

For a right module $M_R$, a left module ${}_{R}N$, and an abelian group $P$, every balanced map $b:M\times N\to P$ factors uniquely through an additive homomorphismExistence follows because additivity and the balancing equation are exactly the relations imposed on the generating symbols $m\otimes n$. Uniqueness follows because those symbols generate the tensor product as an abelian group.

For modules over a commutative ring, if $b$ is $R$-bilinear and the target is an $R$-module, the induced map is an R-module homomorphism. This is the linear version of the universal property; the additive version alone must not silently impose an action on the target.