von Neumann regular
An element of a ring is said to be von Neumann regular if there exists such that . Such an element is known as a of .
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.
A ring is said to be a von Neumann regular ring (or simply a regular ring, if the is clear from context) if every element of is von Neumann regular.
For example, any division ring is von Neumann regular, and so is any ring of matrices over a division ring. In general, any semisimple ring 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 |
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Canonical name | VonNeumannRegular |
Date of creation | 2013-03-22 12:56:18 |
Last modified on | 2013-03-22 12:56:18 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 13 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 16E50 |
Defines | von Neumann regular ring |
Defines | regular ring |
Defines | pseudoinverse |