Universal property of the tensor product

The universal property of the tensor product says that for a right $R$-module $M$ and a left $R$-module $N$, a pair $(T,\tau)$ is a tensor product if $\tau\colon M\times N\to T$ is a balanced bilinear map and for every abelian group $A$ and every balanced bilinear map $f\colon M\times N\to A$, there exists a unique group homomorphism $\overline f\colon T\to A$ such that $\overline f\circ\tau=f$.

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).

Examples:

• The canonical pairing $M\times R\to M$, $(m,r)\mapsto mr$, induces a natural isomorphism $M\otimes_R R\cong M$.
• Any balanced bilinear pairing $M\times N\to P$ factors uniquely as $M\times N\to M\otimes_R N\to P$.