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

Definition

Let $R$ be a commutative ring with unity.

Let $I$ be a set.

Let $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ 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 $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ if and only if 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 $S$.

Also see

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