# Property: von Neumann regular

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

• 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

## Metaproperties:

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