Module on Cartesian Product is Module
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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 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 $26.1$