Tensor product of modules

The tensor product of modules of a right $R$-module $M$ and a left $R$-module $N$ (for a ring $R$) is an abelian group $M\otimes_R N$ equipped with a canonical balanced bilinear map $M\times N\to M\otimes_R N$, $(m,n)\mapsto m\otimes n$, satisfying the universal property : every balanced bilinear map out of $M\times N$ factors uniquely through a homomorphism out of $M\otimes_R N$.

This construction is functorial in both variables and is central for “extension of scalars” and for measuring non-exactness via derived functors. Over commutative rings it specializes to the tensor product of left modules .

Examples:

• For abelian groups (i.e. $\mathbb Z$-modules), $\mathbb Z/n\mathbb Z\otimes_{\mathbb Z}\mathbb Z/m\mathbb Z \cong \mathbb Z/\gcd(n,m)\mathbb Z$.
• For a field $k$ and finite-dimensional vector spaces $V,W$, one has $\dim_k(V\otimes_k W)=\dim_k(V)\dim_k(W)$.
• If $R$ is commutative, $I$ is an ideal of $R$, and $M$ is an $R$-module, then $(R/I)\otimes_R M \cong M/IM$ as a quotient module .