Unitary Module of All Mappings is Unitary Module

Theorem

Let $\struct {R, +_R, \times_R}$ be a ring with unity whose unity is $1_R$.

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.

Let $S$ be a set.

Let $\struct {G^S, +_G', \circ}_R$ be the module of all mappings from $S$ to $G$.

Then $\struct {G^S, +_G', \circ}_R$ is a unitary module.

From Module of All Mappings is Module, we have that $\struct {G^S, +_G', \circ}_R$ is an $R$-module.

To show that $\struct {G^S, +_G', \circ}_R$ is a unitary $R$-module, we verify the following:

$\forall f \in G^S: 1_R \circ f = f$

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.

Then:

$\ds \forall x \in G: \, $

$\ds 1_R \circ x$

$=$

$\ds x$

Left Module Axiom $\text M 4$: for $\struct {G, +_G, \circ}_R$

Thus:

$\ds \forall f \in G^S, \forall x \in G: \, $

$\ds \map {\paren {1_R \circ f} } x$

$=$

$\ds 1_R \circ \paren {\map f x}$

$\ds $

$=$

$\ds \map f x$

Definition of Unity of Ring

Hence the result.

$\blacksquare$

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.4$