---
tag: GVN1MS9PEJ
kind: definition
name: Prime ideal in a noncommutative ring
msc: [16N60, 16D25]
defines: E8K2MN5V1E
uses: [0894J980RK, 0DDA6CY7GJ]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_ideal
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
A [[two-sided ideal|0DDA6CY7GJ]] $P$ of $R$ is *prime* if the following hold.

1. $P \ne R$.
2. For all two-sided ideals $A, B$ of $R$, if $AB \subseteq P$, then $A \subseteq P$ or $B \subseteq P$.

Here $AB$ is the ideal generated by the products $ab$ with $a \in A$ and $b \in B$.

## Remarks

Equivalently, $P \ne R$ and, for all $a, b \in R$, $aRb \subseteq P$ implies $a \in P$ or $b \in P$; and $P$ is prime if and only if $R/P$ is a [[prime ring|Z9TJSZZYG1]].
Example: the zero ideal of $M_2(\Q)$ is prime, since the only two-sided ideals are $0$ and $M_2(\Q)$, and $M_2(\Q) \cdot M_2(\Q) \ne 0$.
Non-example: the zero ideal of $\Q \times \Q$ fails clause 2 with $A = \Q \times 0$ and $B = 0 \times \Q$, since $AB = 0$.
A *completely prime* ideal is one with $ab \in P \Rightarrow a \in P$ or $b \in P$; it is prime, but the zero ideal of $M_2(\Q)$ shows the converse fails.
For a commutative ring both notions agree with the usual [[prime ideal|00MHHZ9SNE]].