Module on Cartesian Product of Ring with Unity is Unitary Module

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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 1

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

$\blacksquare$


Proof 2

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

$\blacksquare$


Proof 3

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

$\blacksquare$


Sources