Module on Cartesian Product is Module

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

Let $n \in \N_{>0}$.

Let $\struct {R^n, +, \times}_R$ be the $R$-module $R^n$.

Then $\struct {R^n, +, \times}_R$ is an $R$-module.