Direct Product of Modules is Module

Theorem
Let $R$ be a ring.

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

Let $\left\langle{M, +, \circ}\right\rangle$ be their direct product.

Then $\left\langle{M, +, \circ}\right\rangle$ is a module.

Proof
From External Direct Product of Abelian Groups is Abelian Group it follows that $(M,+)$ is an abelian group.

Also see

 * Direct Product of Unitary Modules is Unitary Module