Module of All Mappings is Module

Theorem
Let $$R$$ be a ring.

Let $$\left({G, +_G: \circ}\right)_R$$ be an $R$-module.

Let $$S$$ be a set.

Let $$G^S$$ be the set of all mappings from $$S$$ to $$G$$.

Then $$\left({G^S, +_G': \circ}\right)_R$$ is an $R$-module, where:
 * $$+_G'$$ is the operation induced on $$G^S$$ by $$+_G$$;
 * $$\forall \lambda \in R: \forall f \in G^S: \forall x \in S: \left({\lambda \circ f}\right) \left({x}\right) = \lambda \circ f \left({x}\right)$$.

This is the $R$-module $$G^S$$ of all mappings from $$S$$ to $$G$$.

If $$\left({G, +_G: \circ}\right)_R$$ is a unitary $R$-module, then $$\left({G^S, +_G': \circ}\right)_R$$ is also unitary.

The most important case of this example is when $$\left({G^S, +_G': \circ}\right)_R$$ is the $R$-module $$\left({R^S, +_R': \circ}\right)_R$$.