---
tag: 6ZBNGS0P98
kind: concept
name: Homogeneous ideal
msc: [13A02, 13A15]
specializes: [096Z9MP2ME]
sources:
  - url: https://en.wikipedia.org/wiki/Graded_ring
---

## Summary

A homogeneous ideal of a [[graded ring|8KB4SQGC0C]] is an [[ideal|096Z9MP2ME]] that contains all homogeneous components of each of its elements.
Equivalently, it is generated by homogeneous elements.
Homogeneous ideals of a polynomial ring are the ideals that define subsets of projective space.
The quotient by a homogeneous ideal is again graded.

## Explanations

### L1 human

A polynomial is a sum of homogeneous parts of different degrees.
An ideal is homogeneous when it never mixes degrees: whenever it contains a polynomial, it contains each homogeneous part separately.
The ideal generated by $x^2 - yz$ in $k[x, y, z]$ is homogeneous, while the ideal generated by $x - 1$ in $k[x]$ is not, since it contains $x - 1$ but not $x$.
The [[definition|3PNJYFPJ3P]] makes this precise.