---
tag: 06HC6HK0Z8
kind: definition
name: Prime ideal
msc: [13A15]
defines: 00MHHZ9SNE
uses: [0MYSA6MYNA, 0GYGHKX834]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_ideal
    locator: Definition
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An [[ideal|0MYSA6MYNA]] $P$ of $R$ is prime if it has these two properties.

1. $P \neq R$.
2. For all $a, b \in R$, if $a b \in P$, then $a \in P$ or $b \in P$.

## Remarks

In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 2$ is prime if and only if $n$ is a prime number. The zero ideal of $\mathbb{Z}$ is also prime.
The zero ideal of $R$ is prime if and only if $R$ is an [[integral domain|09S3PFBEPM]].
An ideal $P$ is prime if and only if [[$R/P$ is an integral domain|0AVXBEC05W]].
Every [[maximal ideal|09J6KGT65G]] is prime.

## Explanations

### L1 human

A prime number $p$ has this property: if $p$ divides $a b$, then $p$ divides $a$ or $b$.
A prime ideal has the same property, with "divides" replaced by "contains".
The ideal $(6)$ is not prime: it contains $2 \cdot 3$ but neither $2$ nor $3$.