von Neumann regular ring

Summary

A von Neumann regular ring is a ring in which every element $a$ has a weak inverse $x$, with $a x a = a$. Fields, division rings, matrix rings over division rings, products of fields and Boolean rings are examples. John von Neumann introduced these rings in his work on continuous geometry. In the commutative case they are exactly the reduced rings of Krull dimension $0$, and also the rings over which every module is flat. The name is unrelated to regular local rings.