Von Neumann Regular Rings

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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



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].


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

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