---
tag: 3PNJYFPJ3P
kind: definition
name: Homogeneous ideal
msc: [13A02, 13A15]
defines: 6ZBNGS0P98
uses: [8KB4SQGC0C, 096Z9MP2ME, 0GSW76Q14N]
sources:
  - url: https://en.wikipedia.org/wiki/Graded_ring
---

## Definition

Let $R = \bigoplus_{n \ge 0} R_n$ be a [[graded ring|8KB4SQGC0C]].
An [[ideal|096Z9MP2ME]] $I$ of $R$ is *homogeneous* if for every $f \in I$ with homogeneous components $f_n \in R_n$, each $f_n$ lies in $I$.

## Remarks

Equivalent conditions: $I = \bigoplus_{n \ge 0} (I \cap R_n)$; and $I$ is [[generated|0GSW76Q14N]] by homogeneous elements.
Example: in $k[x, y, z]$, with $k$ a field, the ideal $(x^2 - yz)$ is homogeneous, since it is generated by the homogeneous element $x^2 - yz$.
Non-example: in $k[x]$ the ideal $(x - 1)$ is not homogeneous; it contains $x - 1$, whose component $-1$ of degree $0$ is a unit and does not lie in the proper ideal $(x - 1)$.
If $I$ is homogeneous, then $R/I$ is graded by the images of the $R_n$.
The ideal $R_+ = \bigoplus_{n \ge 1} R_n$ is homogeneous; it is called the irrelevant ideal.