Definition:Module Direct Product

Definition
Let $\left\{ {M_i}\right\}_{i \in I}$ be a family of $R$-left modules.

Let $M = \displaystyle \prod_{i \mathop \in I} M_i$ be the cartesian product of all these modules.

Define the operation $+$ on $M$ with:
 * $a_i, b_i \in M_i$ and $\left\langle{a_i}\right\rangle_{i \mathop \in I}, \left\langle{b_i}\right\rangle_{i \mathop \in I} \in M$

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

Also define the $R$-action $\circ$ on $M$ as:


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

Then $\left\langle{M, \circ}\right\rangle$ is a direct product of left modules

Also see

 * Universal Property of Direct Product of Modules
 * Definition:Module Direct Sum