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 $\left({M, +, \circ}\right)$ be their direct product.

Then $\left({M, +, \circ}\right)$ 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: