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 Induced Group and Induced Structure Associative, we have that $$\left({R^S, +'}\right)$$ is a group and $$\left({R^S, \circ'}\right)$$ 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.