# 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

## 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