= Tensor product of modules
{title2=$M\otimes_RN$}
{wiki}
For a <right module> $M_R$ and a left <module> ${}_RN$, their <tensor product of modules> is the <abelian group> generated by symbols $m\otimes n$, modulo additivity in each variable and
$$
(mr)\otimes n=m\otimes(rn).
$$
The <universal property of the tensor product of modules> identifies additive maps from it with <balanced maps> on $M\times N$.
When $R$ is a <commutative ring>, two left modules can be used by viewing the first as a right module. The tensor product is then an $R$-module with $r(m\otimes n)=(rm)\otimes n=m\otimes(rn)$. For example, $\mathbb Z/a\mathbb Z\otimes_{\mathbb Z}\mathbb Z/b\mathbb Z\cong\mathbb Z/\gcd(a,b)\mathbb Z$: the generator $1\otimes1$ is killed by both $a$ and $b$, and the universal property identifies exactly those relations. For a noncommutative ring, the left and right input actions must be specified.
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