Unitary Module of All Mappings is Unitary Module

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Theorem

Let $\struct {R, +_R, \times_R}$ be a ring.

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.


Proof

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:

$\forall x \in G: 1_R \circ x = x$


Thus:

\(\displaystyle \map {\paren {1_R \circ f} } x\) \(=\) \(\displaystyle 1_R \circ \paren {\map f x}\)
\(\displaystyle \) \(=\) \(\displaystyle \map f x\)

Hence the result.

$\blacksquare$


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