\begin{definition}[Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1]
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 mapping $a: I \to \N$ with finite support such that:
:$y = \ds \prod_{i \mathop \in I} X_i^{a_i}$
where:
:$\prod$ denotes the product with finite support over $I$
:$X_i^{a_i}$ denotes the $a_i$th power of $X_i$.
\end{definition}