Polynomial Functions form Submodule of All Functions

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

Let $K^K$ be the $K$-module of all mappings $f: K \to K$, as described on Module of All Mappings.

Let $P \left({K}\right) \subseteq K^K$ be the set of all polynomial functions on $K$.

Then $P \left({K}\right)$ is a $K$-submodule of $K^K$.