von Neumann regular
For example, any unit in a ring is von Neumann regular. Also, any idempotent element is von Neumann regular. For a non-unit, non-idempotent von Nuemann regular element, take , the ring of matrices over . Then
is von Neumann regular. In fact, we can replace with any non-zero and the resulting matrix is also von Neumann regular. There are several ways to generalize this example. One way is take a central idempotent in any ring , and any with . Then is von Neumann regular, with and all as pseudoinverses. In another generalization, we have two rings where is an algebra over . Take any idempotent , and any invertible element such that commutes with . Then is von Neumann regular.
Remark. Note that regular ring in the sense of von Neumann should not be confused with regular ring in the sense of , which is a Noetherian ring whose localization at every prime ideal is a regular local ring.
|Title||von Neumann regular|
|Date of creation||2013-03-22 12:56:18|
|Last modified on||2013-03-22 12:56:18|
|Last modified by||CWoo (3771)|
|Defines||von Neumann regular ring|