Definition:Monomial of Polynomial Ring

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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 $S$.


Also see