Definition:Tensor Product of Modules

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Definition

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 $\tuple {m, r n}$

Let:

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

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 $b \in N$.


Notes


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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


Sources


This article, or a section of it, needs explaining.
In particular: It is useful to specify exactly where in the source works the citation comes. Also, it is necessary to be precise about the editions used. Note that Lang's "Algebra" came out in 1965, and Bourbaki's work originally dates from the 1940's. The 2002 date suggests that the Lang work being used here is the 3rd edition. Haven't a clue about the Bourbaki, I find them unreadable and so have not paid much attention to them.
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