Definition:Module Direct Product

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Definition

Finite Case

Let $R$ be a ring.

Let $\left({M_1, +_1, \circ_1}\right)_R, \left({M_2, +_2, \circ_2}\right)_R, \ldots, \left({M_n, +_n, \circ_n}\right)_R$ be $R$-modules.

Let:

$\displaystyle M = \prod_{k \mathop = 1}^n M_k$

be the cartesian product of $M_1$ to $M_n$.

Let:

$+$ be the operation induced on $M$ by the operations $+_1, +_2, \ldots, +_n$ on $M_1, M_2, \ldots, M_n$
$\circ$ be defined as $\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) = \left({\lambda \circ_1 x_1, \lambda \circ_2 x_2, \ldots, \lambda \circ_n x_n}\right)$

In Finite Direct Product of Modules is Module, it is shown that $\left\langle{M, +, \circ}\right\rangle$ is an $R$-module.

The module $\left({M, +, \circ}\right)_R$ is called the (external) direct product of $M_1$ to $M_n$.


General Case

Let $R$ be a ring.

Let $\left\{ \left\langle M_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$ be a family of $R$-modules.



Let:

$M = \displaystyle \prod_{i \mathop \in I} M_i$

be the cartesian product of these modules.


The operation $+$ induced on $M$ by $(+_i)_{i\in I}$ is the operation defined by:

$\left\langle{a_i}\right\rangle_{i \mathop \in I} + \left\langle{b_i}\right\rangle_{i \mathop \in I} = \left\langle{a_i +_i b_i}\right\rangle_{i \mathop \in I}$


That is, the additive group of the module $M$ is the direct product of the groups $\left\{ \left(M_i,+_i\right)\right\}_{i \in I}$.


The $R$-action $\circ$ induced on $M$ by $(\circ_i)_{i\in I}$ is the operation defined by:

$r \circ \left\langle{m_i}\right\rangle_{i \mathop \in I} = \left\langle{r \circ_i m_i}\right\rangle_{i \mathop \in I}$

In Direct Product of Modules is Module, it is shown that $\left\langle{M, +, \circ}\right\rangle$ is an $R$-module.


The module $\left\langle{M, +, \circ}\right\rangle$ is called the (external) direct product of $\left\{ \left\langle M_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$.


Also see