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The Nilpotent Regular Element Problem

  Published:2016-05-10
 Printed: Sep 2016
  • Pere Ara,
    Department of Mathematics, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barċelona), Spain
  • Kevin C. O'Meara,
    2901 Gough Street, Apartment 302, San Francisco, CA 94123, USA
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Abstract

We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.
Keywords: nilpotent element, von Neumann regular element, unit-regular, Bergman's normal form nilpotent element, von Neumann regular element, unit-regular, Bergman's normal form
MSC Classifications: 16E50, 16U99, 16S10, 16S15 show english descriptions von Neumann regular rings and generalizations
None of the above, but in this section
Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16E50 - von Neumann regular rings and generalizations
16U99 - None of the above, but in this section
16S10 - Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16S15 - Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
 

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