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 $$K^K$$.

Consider the set $$P_m \left({K}\right)$$ of all the polynomial functions:
 * $$\sum_{k=0}^{m-1} \alpha_k {I_K}^k$$

for some $$m \in \N^*$$ where:
 * $$\left \langle {\alpha_k} \right \rangle_{k \in \left[{0 \, . \, . \, m-1}\right]}$$

is any sequence of $$m$$ terms of $$K$$.

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