---
tag: 00MHHZ9SNE
kind: concept
name: Prime ideal
msc: [13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_ideal
    locator: lead section
same_as:
  wikidata: Q863912
  wikipedia: Prime ideal
---

## Summary

A prime ideal of a commutative ring is a proper ideal with the following property. If it contains a product, then it contains one of the factors.
The prime ideals of $\mathbb{Z}$ are the ideals $p\mathbb{Z}$ for $p$ prime and the zero ideal.
The [[definition of a prime ideal|06HC6HK0Z8]] gives the conditions.