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 $P \left({K}\right)$.