Von Neumann regular ring

In mathematics, a von Neumann regular ring is a ring R (associative, with 1, not necessarily commutative) such that for every element a in R there exists an x in R with a = axa. One may think of x as a "weak inverse" of the element a; in general x is not uniquely determined by a. Von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left R-module is flat.

Von Neumann regular rings were introduced by von Neumann (1936) under the name of "regular rings", in the course of his study of von Neumann algebras and continuous geometry. Von Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra.

An element a of a ring is called a von Neumann regular element if there exists an x such that a = axa.^[1] An ideal ${\displaystyle {\mathfrak {i}}}$ is called a (von Neumann) regular ideal if for every element a in ${\displaystyle {\mathfrak {i}}}$ there exists an element x in ${\displaystyle {\mathfrak {i}}}$ such that a = axa.^[2]