Universal property of the tensor product

Let $R$ be a ring, $M$ a right $R$-module, and $N$ a left $R$-module. A pair $(T,\tau)$ has the universal property of the tensor product if $T$ is an abelian group, $\tau\colon M\times N\to T$ is biadditive and balanced,

and every biadditive balanced map $f\colon M\times N\to A$ to an abelian group $A$ factors uniquely through a homomorphism $\overline f:T\to A$.

When such $(T,\tau)$ exists, it is unique up to unique isomorphism; one writes $T=M\otimes_R N$ and $\tau(m,n)=m\otimes n$, producing the tensor product. The universal property is the mechanism that turns bilinear constructions into linear ones (i.e. module homomorphisms, when the target has compatible structure).