Tensor product universal property

Let $M$ be a right $R$-module and $N$ a left $R$-module. There is an abelian group $M\otimes_R N$ and a biadditive map

satisfying $\tau(mr,n)=\tau(m,rn)$. For every abelian group $A$ and every biadditive map $b:M\times N\to A$ satisfying $b(mr,n)=b(m,rn)$, there is a unique group homomorphism $\phi:M\otimes_R N\to A$ such that $b=\phi\circ\tau$.

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