Submodule of Module of Polynomial Functions

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Let $K$ be a commutative ring with unity.

Let $\map P K$ be the set of all polynomial functions on $K$.

Consider the set $\map {P_m} K$ of all the polynomial functions:

$\ds \sum_{k \mathop = 0}^{m - 1} \alpha_k {I_K}^k$

for some $m \in \N^*$ where:

$\sequence {\alpha_k}_{k \in \closedint 0 {m - 1} }$

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

Then $\map {P_m} K$ is a submodule of $\map P K$.