Definition:Tensor Product of Modules

Work here, pay no attention to the man behind the curtains -- Let $R$ be a given ring, $M$ a right module and $N$ a left module. Construct first a module of all possible ordered pairs.

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

Next for all $m,m'\in M$, $n,n'\in N$ and $r\in R$ we define a set $\mathbb{D}$ as set of all the elements Next we construct this module $$D = \bigoplus_{d\in\mathbb{D}} R d$$
 * $(m+m',n)-(m,n)-(m',n)$
 * $(m,n+n')-(m,n)-(m,n')$
 * $r(m,n)-(mr,n)$
 * $r(m,n)-(m,rn)$

The tensor product $M\otimes N$ is then our quotient module $T/D$