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 modules

Define the operation $+$ on $M$ with $a_i,b_i\in M_i$ and $(a_i)_{i\in I},(a_i)_{i\in I}\in M$ as

$(a_i)_{i\in I}+(b_i)_{i\in I}=(a_i+b_i)_{i\in I}$

Also Define the $R$-action $\circ$ on $M$ as

$r\circ(m_i)_{i\in I}=(r\circ_i m_i)_{i\in I}$

Then $(M,\circ)$ is a direct product of left modules

Also see

 * Definition:Module Direct Sum