Definition:Polynomial over Ring as Function on Free Monoid on Set

Definition
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.

Also see

 * Definition:Polynomial over Ring as Sequence