Definition:Tensor Product of Modules

Definition
Let $\mathbf{R}$ be a given ring, $M$ a right module and $N$ a left module. Construct first a module of all possible ordered pairs.

$T = \displaystyle\bigoplus_{s\in M\times N} R s$

This satesfies all module axioms. Next for all $m,m'\in M$, $n,n'\in N$ and $r\in \mathbf{R}$ define a set $\mathbb{D}$ as set of all the elements Let
 * $(m+m',n)-(m,n)-(m',n)$
 * $(m,n+n')-(m,n)-(m,n')$
 * $r(m,n)-(mr,n)$
 * $r(m,n)-(m,rn)$

$D = \displaystyle\bigoplus_{d\in\mathbb{D}} \mathbf{R} d$

The tensor product $M\otimes_\mathbf{R} N$ is then our quotient module $T/D$