Definition:Monomial of Polynomial Ring
Definition
Let $R$ be a commutative ring with unity.
One Variable
Let $R \sqbrk X$ be a polynomial ring over $R$ in one indeterminate $X$.
A monomial of $R \sqbrk X$ is an element that is a power of $X$.
Multiple Variables
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 $y \in R \sqbrk {\family {X_i}_{i \mathop \in I} }$.
A monomial of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ is an element that is a product of variables; specifically:
Definition 1
The element $y$ is a monomial of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ if and only if 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 $X_i$.
Definition 2
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 $S$.