Property: von Neumann regular

Definition: For every $x$, there exists $y$ such that $x=xyx$

Resources for learning about this property:

  • A. A. Tuganbaev. Rings close to regular. (2013) @ Chapter 2
  • K. R. Goodearl. Von Neumann regular rings. (1991) @ .
  • T.-Y. Lam. Lectures on modules and rings. (2012) @ Section 4B


This property has the following metaproperties
  • passes to $eRe$ for any idempotent $e$
  • passes to $eRe$ for any full idempotent $e$
  • Morita invariant
  • passes to matrix rings
  • passes to the center
This property does not have the following metaproperties