Definition:Module Direct Product/General Case

Definition

Let $R$ be a ring.

Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of $R$-modules.

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Let:

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

be the cartesian product of these modules.

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

$\family {a_i}_{i \mathop \in I} + \family {b_i}_{i \mathop \in I} = \family {a_i +_i b_i}_{i \mathop \in I}$

That is, the additive group of the module $M$ is the direct product of the groups $\family {\struct {M_i, +_i} }_{i \mathop \in I}$.

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

$r \circ \family {m_i}_{i \mathop \in I} = \family {r \circ_i m_i}_{i \mathop \in I}$

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

The module $\struct {M, +, \circ}$ is called the (external) direct product of $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$.