Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 2

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

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 $S$.

Also see

 * Equivalence of Definitions of Monomial of Polynomial Ring in Multiple Variables