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.

Proof 1

This is a special case of Direct Product of Modules is Module.

$\blacksquare$

Proof 2

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

$\blacksquare$

Proof 3

This is a special case of a Finite Direct Product of Modules is Module where each of the $G_k$ is the $R$-module $R$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.1$