Polynomial Functions form Submodule of All Functions

Theorem
Let $K$ be a commutative ring with unity.

Let $K^K$ be the $K$-module mappings $f: K \to K$.

Let $\map P K \subseteq K^K$ be the set of all polynomial functions on $K$.

Then $\map P K$ is a $K$-submodule of $K^K$.