von Neumann regular

An element a of a ring R is said to be von Neumann regular if there exists bR such that aba=a. Such an element b is known as a of a.

For example, any unit in a ring is von Neumann regular. Also, any idempotent element is von Neumann regular. For a non-unit, non-idempotent von Nuemann regular elementPlanetmathPlanetmath, take M2(), the ring of 2×2 matrices over . Then


is von Neumann regular. In fact, we can replace 2 with any non-zero r and the resulting matrix is also von Neumann regular. There are several ways to generalize this example. One way is take a central idempotent e in any ring R, and any rs=f with ef=e. Then re is von Neumann regular, with s,se and sf all as pseudoinverses. In another generalizationPlanetmathPlanetmath, we have two rings R,S where R is an algebra over S. Take any idempotent eR, and any invertible element sS such that s commutes with e. Then se is von Neumann regular.

A ring R is said to be a von Neumann regular ring (or simply a regular ring, if the is clear from context) if every element of R is von Neumann regular.

For example, any division ring is von Neumann regular, and so is any ring of matrices over a division ring. In general, any semisimple ringPlanetmathPlanetmath is von Neumann regular.

Remark. Note that regular ring in the sense of von Neumann should not be confused with regular ring in the sense of , which is a Noetherian ringMathworldPlanetmath whose localizationMathworldPlanetmath at every prime idealMathworldPlanetmathPlanetmathPlanetmath is a regular local ringMathworldPlanetmath.

Title von Neumann regular
Canonical name VonNeumannRegular
Date of creation 2013-03-22 12:56:18
Last modified on 2013-03-22 12:56:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 16E50
Defines von Neumann regular ring
Defines regular ring
Defines pseudoinverse