11562

# 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.

### DETAILS

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].
Reference
[1] K. R. Goodearl, Von Neumann Regular Rings, San Francisco: Pitman, 1979.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.