---
tag: Z9TJSZZYG1
kind: concept
name: Prime ring
msc: [16N60]
specializes: [5F0YMCV50R]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_ring
same_as:
  wikidata: Q386597
  wikipedia: Prime ring
---

## Summary

A prime ring is a nonzero ring in which $aRb = 0$ forces $a = 0$ or $b = 0$.
Equivalently, the product of two nonzero two-sided ideals is nonzero, that is, the zero ideal is a [[prime ideal|E8K2MN5V1E]].
It is the noncommutative replacement for an integral domain: a commutative ring is prime exactly when it is an integral domain.
Matrix rings over prime rings are prime, so prime rings may have many zero divisors.

## Explanations

### L1 human

In an integral domain, $ab = 0$ forces $a = 0$ or $b = 0$.
Matrix rings fail this, since two nonzero matrices can multiply to zero.
A prime ring asks for something weaker: $a$ times anything times $b$ must not always be zero.
The ring $M_2(\Q)$ is prime even though $e_{11} e_{22} = 0$, because $e_{11} e_{12} e_{22} = e_{12} \ne 0$.