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von Neumann regular
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(Definition)
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An element  of a ring  is said to be von Neumann regular if there exists  such that  . Such an element  is known as a pseudoinverse 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 meaning 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 commutative algebra, which is a Noetherian ring whose localization at every prime ideal is a regular local ring.
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"von Neumann regular" is owned by CWoo. [ full author list (2) | owner history (1) ]
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(view preamble)
| Also defines: |
von Neumann regular ring, regular ring, pseudoinverse |
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Cross-references: regular local ring, prime ideal, localization, noetherian ring, semisimple ring, division ring, clear, invertible, algebra, matrices, regular element, idempotent element, unit, ring
There are 5 references to this entry.
This is version 10 of von Neumann regular, born on 2002-08-14, modified 2007-07-25.
Object id is 3295, canonical name is VonNeumannRegular.
Accessed 4629 times total.
Classification:
| AMS MSC: | 16E50 (Associative rings and algebras :: Homological methods :: von Neumann regular rings and generalizations) |
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Pending Errata and Addenda
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