Polynomial Functions form Submodule of All Functions

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

Let the mapping $$p: K \to K$$ be defined such that there exists a sequence $$\left \langle {\alpha_k} \right \rangle_{k \in \left[{0 \,. \, . \, n}\right]}$$ of elements of $$K$$ such that $$p = \sum_{k=0}^n \alpha_k {I_K}^k$$

where $$I_K$$ is the identity mapping on $$K$$.

Then $$p$$ is known as a polynomial function on $$K$$.

The set $$P \left({K}\right)$$ of all polynomial functions on $$K$$ is a submodule of the $K$-module $$K^K$$.

If $$m \in \mathbb{N}^*$$, then the set $$P_m \left({K}\right)$$ of all the polynomial functions $$\sum_{k=0}^{m-1} \alpha_k {I_K}^k$$

where $$\left \langle {\alpha_k} \right \rangle_{k \in \left[{0 \,. \, . \, m-1}\right]}$$ is any sequence of $$m$$ terms of $$K$$, is a submodule of $$P \left({K}\right)$$.