---
tag: Z7MTTS6FFG
kind: definition
name: Monomial ideal
msc: [13F55, 13F20, "05E40"]
defines: 4G25S7PJ22
uses: [04C5ZBG1WT, 0SDC934R52, 096Z9MP2ME]
sources:
  - url: https://en.wikipedia.org/wiki/Monomial_ideal
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]].
A *monomial* of $S$ is a product $x^a = x_1^{a_1} \cdots x_n^{a_n}$ with $a = (a_1, \ldots, a_n) \in \N^n$.
An [[ideal|096Z9MP2ME]] $I$ of $S$ is a *monomial ideal* if it is generated by monomials.

## Remarks

An equivalent form: $I$ is a monomial ideal if and only if, for every $f \in I$, each monomial that occurs in $f$ with a nonzero coefficient lies in $I$.
A monomial $x^b$ lies in the monomial ideal generated by monomials $x^{a}$, $a \in A$, if and only if $b - a \in \N^n$ for some $a \in A$.
By Dickson's lemma, every monomial ideal is generated by finitely many monomials, and it has a unique minimal set of monomial generators.

Example: $I = (x^2, xy)$ in $k[x, y]$ is a monomial ideal by definition.
The polynomial $x^3 + x y^5$ lies in $I$, and indeed both of its monomials are multiples of $x^2$ or of $xy$.

Non-example: $I = (x + y)$ in $k[x, y]$ is not a monomial ideal.
It contains $x + y$, but it does not contain the monomial $x$, since every nonzero element of $I$ is a multiple of $x + y$.