Finite Direct Product of Unitary Modules is Unitary Module

Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be unitary $R$-modules.

Let:
 * $\displaystyle G = \prod_{k \mathop = 1}^n G_k$

be their direct product.

Then $G$ is a unitary module.