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.

First construct a left module as a direct sum of all free left modules with a basis that is a single ordered pair in $M \times N$ which is denoted $R \left({m, n}\right)$.


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

That this is indeed a module is demonstrated in Tensor Product is Module.

Next for all $m, m' \in M$, $n, n' \in N$ and $r \in R$ we construct the following free left modules.


 * $L_{m, m', n}$ with a basis of $\left({m + m', n}\right)$, $\left({m, n}\right)$ and $\left({m', n}\right)$
 * $R_{m, n, n'}$ with a basis of $\left({m, n + n'}\right)$, $\left({m, n}\right)$ and $\left({m, n'}\right)$
 * $A_{r, m, n}$ with a basis of $r \left({m, n}\right)$ and $\left({m r, n}\right)$
 * $B_{r, m, n}$ with a basis of $r \left({m, n}\right)$ and $\left({m, r n}\right)$

Let:


 * $D = \displaystyle \bigoplus_{r\in R, n,n'\in N, m,m'\in M} (L_{m,m',n}\oplus R_{m,n,n'} \oplus A_{r,m,n} \oplus B_{r,m,n})$

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

Also see

 * Tensor Product is Module
 * Tensor of zero is zero in tensor