Definition:Monomial of Polynomial Ring/Multiple Variables

Definition
Let $R$ be a commutative ring with unity. Let $I$ be a set.

Let $R[(X_i)_{i\in I}]$ be a polynomial ring in $I$ variables $(X_i)_{i\in I}$.

Let $y \in R[(X_i)_{i\in I}]$.

A monomial of $R[(X_i)_{i\in I}]$ is an element that is a product of variables; specifically:

Definition 1
The element $y$ is a monomial of $R[(X_i)_{i\in I}]$ there exists a mapping $a : I \to \N$ with finite support such that:
 * $y = \displaystyle \prod_{i \in I} X_i^{a_i}$

where:
 * $\prod$ denotes the product with finite support over $I$
 * $X_i^{a_i}$ denotes the $a_i$th power of $X_i$.

Definition 2
The element $y$ is a monomial of $R[(X_i)_{i\in I}]$ there exists a finite set $S$ and a mapping $f : S \to \{X_i : i\in I\}$ such that it equals
 * $y = \displaystyle \prod_{s \in S} f(s)$

where $\prod$ denotes the product over the finite set $S$.