Module on Cartesian Product of Ring with Unity is Unitary Module/Proof 2

Theorem

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

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

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

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

Proof

This is a special case of the Unitary Module of All Mappings is Unitary Module, where $S$ is the set $\closedint 1 n \subset \N_{>0}$.

$\blacksquare$