Von Neumann Regular Rings
This Demonstration shows that the algebra of matrices over the rational numbers is a von Neumann regular ring. A ring is von Neumann regular provided that for every there exists such that . Every field is a regular ring, since , if . The simplest nontrivial example of a regular ring is an algebra of matrices over some field. If , then , so is an idempotent in . Similarly, is an idempotent. In Wolfram Mathematica, we get from by using the built-in PseudoInverse function.
Nontrivial examples for are matrices of rank 1. Obviously, nonzero matrices with one row or column of zeros are natural choices.
It can be shown that matrices over a regular ring form a regular ring [1, pp. 4–7].
 K. R. Goodearl, Von Neumann Regular Rings, San Francisco: Pitman, 1979.