Definition:Module Direct Product

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

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

Define the operation $+$ on $M$ 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 modules