\begin{definition}[Definition:Polynomial over Ring as Function on Free Monoid on Set]
Let  $R$ be a  commutative ring with unity.
Let $M$ be the free commutative monoid on the indexed set $\left\{{X_j: j \in J}\right\}$.
A '''polynomial''' in $\left\{{X_j: j \in J}\right\}$ can be defined as a mapping $f: M \to R$ of finite support.
That is, it is an element of the ring of polynomial forms.
\end{definition}