Definition:Tensor Product of Modules

Commutative ring
Let $R$ be a commutative ring with unity.

Let $M$ and $N$ be $R$-modules.

Definition 1
Their tensor product is a pair $(M \otimes_R N, \theta)$ where: satisfying the following universal property:
 * $M \otimes_R N$ is an $R$-module
 * $\theta : M \times N \to M \otimes_R N$ is an $R$-bilinear mapping
 * For every pair $(P, \omega)$ of an $R$-module and an $R$-bilinear mapping $\omega : M \times N \to P$, there exists a unique $R$-module homomorphism $f : M \otimes_R N \to P$ with $\omega = f \circ \theta$.

Definition 2
Their tensor product is the pair $(M \otimes_R N, \theta)$, where:
 * $M \otimes_R N$ is the quotient of the free $R$-module $R^{(M\times N)}$ on the direct product $M \times N$, by the submodule generated by the set of elements of the form:
 * $(\lambda m_1 + m_2, n) - \lambda (m_1, n) - (m_2, n)$
 * $(m, \lambda n_1 + n_2) - \lambda (m, n_1) - (m, n_2)$
 * for $m, m_1, m_2 \in M$, $n, n_1, n_2 \in N$ and $\lambda \in R$, where we denote $(m, n)$ for its image under the canonical mapping $M \times N \to R^{(M\times N)}$.
 * $\theta : M \times N \to M \otimes_R N$ is the composition of the canonical mapping $M \times N \to R^{(M\times N)}$ with the quotient module homomorphism $R^{(M\times N)} \to M \otimes_R N$.

Noncommutative ring
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 denoted as
Elements in $M \otimes N$ are commonly written as $a \otimes b$ for $a \in M$ and $b \in N$.

Also see

 * Definition:Tensor Product of Abelian Groups