Module on Cartesian Product is Module/Proof 2
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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.
Proof
This is a special case of the Module of All Mappings, where $S$ is the set $\closedint 1 n \subset \N_{>0}$.
$\blacksquare$