Submodule of Module of Polynomial Functions

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

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

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

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