# Definition:Monomial of Polynomial Ring

## Contents

## 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 $(X_i)_{i\in I}$.

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:

- $y = \displaystyle \prod_{i \mathop \in I} X_i^{a_i}$

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

- $y = \displaystyle \prod_{s \mathop \in S} \map f s$

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