---
tag: 7PDWB550C9
kind: concept
name: Irreducible ideal
msc: ["13A15"]
specializes: [096Z9MP2ME]
sources:
  - url: https://en.wikipedia.org/wiki/Irreducible_ideal
same_as:
  wikidata: Q17098198
  wikipedia: Irreducible ideal
---

## Summary

An irreducible ideal is a proper ideal that is not the intersection of two strictly larger ideals.
In a [[Noetherian ring|0670T59HGE]], every proper ideal is a finite intersection of irreducible ideals, and every irreducible ideal is [[primary|TMTPWVETQK]].
This is the usual route to the existence part of the [[Lasker–Noether theorem|5H8Z56KXMS]].

## Explanations

### L1 human

An irreducible ideal cannot be split as the overlap of two bigger ideals.
In $\Z$, the ideal $(6)$ splits as $(2) \cap (3)$, so it is not irreducible.
The ideal $(8)$ cannot be split, because every strictly larger ideal is $(4)$, $(2)$ or $\Z$, and these form a chain whose intersections are never $(8)$.