Definition:Tensor Product of Modules

Definition
Let $R$ be a ring.

Let $M$ be a $R$-right module.

Let $N$ be a $R$-left module.

Construct first a module of all possible ordered pairs of elements of $M$ and $N$:


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

This satisfies all module axioms.

Next for all $m, m' \in M$, $n, n' \in N$ and $r \in R$ define a set $\mathbb D$ as set of all the elements:


 * $\left({m + m', n}\right) - \left({m, n}\right) - \left({m', n}\right)$
 * $\left({m, n + n'}\right) - \left({m, n}\right) - \left({m, n'}\right)$
 * $r \left({m, n}\right) - \left({m r, n}\right)$
 * $r \left({m, n}\right) - \left({m, r n}\right)$

Let:


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

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