\begin{definition}[Definition:Module Direct Product/General Case]
Let $R$ be a ring.
Let $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$ be a family of $R$-modules.
{{explain|Expand the notation of the module to specify the operations of the ring.}}
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 $\family {\struct {M_i, +_i, \circ_i} }_{i \mathop \in I}$.
Category:Definitions/Module Theory
Category:Definitions/Direct Products
\end{definition}