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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
 Format: LaTeX MathJax PDF

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
 MSC Classifications: 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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