Direct Product of Unitary Modules is Unitary Module

Theorem
Let $R$ be a ring with unity.

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

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

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

Proof
From Direct Product of Modules is Module, $M$ is an $R$ module.

It remains to verify that
 * $\forall x\in M:$ $1\circ x=x$

We have: