Tensor product universal property

Tensor product universal property: Let $M$ be a right $R$-module and $N$ a left $R$-module. There exists an abelian group $M\otimes_R N$ and an $R$-balanced bilinear map $\tau:M\times N\to M\otimes_R N$ such that for every abelian group $A$ and every $R$-balanced bilinear map $b:M\times N\to A$, there is a unique group homomorphism $\phi:M\otimes_R N\to A$ with $b=\phi\circ \tau$.

This is the standard representing property of the tensor product , packaging bilinear maps into a universal object; compare the tensor product universal property .