\begin{definition}[Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2]
Let $R$ be a commutative ring with unity.
Let $I$ be a set.
Let $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\family {X_i}_{i \mathop \in I}$.
Let $y \in R \sqbrk {\family {X_i}_{i \mathop \in I} }$.
The element $y$ is a '''monomial''' of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ {{iff}} there exists a finite set $S$ and a mapping $f: S \to \set {X_i : i \in I}$ such that it equals
:$y = \ds \prod_{s \mathop \in S} \map f s$
where $\prod$ denotes the product over the finite set $S$.
\end{definition}