Structure Induced by Ring Operations is Ring

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Let $S$ be a set.

Then $\left({R^S, +', \circ'}\right)$ is a ring, where $+'$ and $\circ'$ are the operations induced on $R^S$ by $+$ and $\circ$.

Proof
$\left({R^S, +', \circ'}\right)$ is closed, from the definition of the composition of mappings:


 * $\forall f, g \in R^S: f \circ' g \in R^S$

By Structure Induced by Abelian Group Operation is Abelian Group, $\left({R^S, +'}\right)$ is an abelian group.

By Structure Induced by Associative Operation is Associative $\left({R^S, \circ'}\right)$ is a semigroup.

All that is needed is to show that $\circ'$ is distributive over $+'$.

Let $f, g, h: S \to R$ be elements of $R^S$, the set of all mappings from $S$ to $R$.

Let $x \in S$.

Then:

Similarly we can show:
 * $\left ({\left({g +' h}\right) \circ' f \left({x}\right)}\right) = \left ({\left ({g \circ' f}\right) +' \left ({h \circ' f}\right)}\right) \left({x}\right)$

Hence the result.