Von Neumann Regular Rings

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

Contributed by: Izidor Hafner (February 2018)
Open content licensed under CC BY-NC-SA


Snapshots


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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send