---
tag: DEV24TSWQZ
kind: definition
name: von Neumann regular ring
msc: ["16E50", 13C11]
defines: DXX7GNY3HE
uses: [0B1261B5ED, 0N5NJZKKN3, 04C5ZBG1WT, 0EA5BTA0CX, 0VD9ZSWWJG, 0RCV1ZFP8G, 0QM2G72DXA, SZHY975FFX, MVGBS3J1P7, 73RYYEVDKA]
sources:
  - url: https://en.wikipedia.org/wiki/Von_Neumann_regular_ring
---

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity, not necessarily commutative.
$R$ is *von Neumann regular* if for every $a \in R$ there is $x \in R$ with $a x a = a$.

## Remarks

If $a x a = a$, then $e = a x$ is an [[idempotent|0N5NJZKKN3]] with $e R = a R$; so $R$ is von Neumann regular if and only if every principal right ideal is generated by an idempotent.
A commutative ring is von Neumann regular if and only if it is [[reduced|SZHY975FFX]] of [[Krull dimension|MVGBS3J1P7]] $0$, and if and only if every $R$-module is [[flat|73RYYEVDKA]].
Every semisimple ring, for instance the ring of $n \times n$ [[matrices|0VD9ZSWWJG]] over a [[division ring|0EA5BTA0CX]], is von Neumann regular.

Example: a [[Boolean ring|0RCV1ZFP8G]] $R$, in which $a^2 = a$ for all $a$, is von Neumann regular with $x = a$: $a a a = a^2 = a$.
Example: a [[field|04C5ZBG1WT]] is von Neumann regular, with $x = a^{-1}$ for $a \ne 0$ and $x = 0$ for $a = 0$; so is any [[product|0QM2G72DXA]] of fields, componentwise.

Non-example: $\Z$ is not von Neumann regular.
For $a = 2$, the condition $2 x 2 = 2$ means $4x = 1$, which has no solution in $\Z$.