Definition:Tensor Product of Modules

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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 $\struct {M \otimes_R N, \theta}$ where:

$M \otimes_R N$ is an $R$-module
$\theta : M \times N \to M \otimes_R N$ is an $R$-bilinear mapping

satisfying the following universal property:

For every pair $\struct {P, \omega}$ 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 $\struct {M \otimes_R N, \theta}$, where:

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

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 $\map R {m, n}$.

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

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 $\tuple {m + m', n}$, $\tuple {m, n}$ and $\tuple {m', n}$
$R_{m, n, n'}$ with a basis of $\tuple {m, n + n'}$, $\tuple {m, n}$ and $\tuple {m, n'}$
$A_{r, m, n}$ with a basis of $r \tuple {m, n}$ and $\tuple {m r, n}$
$B_{r, m, n}$ with a basis of $r \tuple {m, n}$ and $\tuple {m, r n}$


$D = \ds \map {\bigoplus_{r \in R, n, n' \in N, m, m' \in M} } {L_{m, m', n} \oplus R_{m, n, n'} \oplus A_{r, m, n} \oplus B_{r, m, n} }$

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 $b \in N$.


This notation, and definition of quotient module, gives us the following identities for $m, m' \in M$, $n, n' \in N$ and $r \in R$:

$\paren {m + m'} \otimes n = m \otimes n + m' \otimes n$
$m \otimes \paren {n + n'} = m \otimes n + m \otimes n'$
$\paren {m r} \otimes n = m \otimes \paren {r n} = r \paren {m \otimes n}$

Note should be taken that not all elements in $M \otimes N$ comes in the simple form of $a \otimes b$, as an example in $\Z \otimes \Z$ we have $2 \otimes 2 + 3 \otimes 5$ being an element in it but cannot be simplified further using the previous identities.

Also see