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Algebras and Representation Theory

, Volume 18, Issue 4, pp 931–940 | Cite as

Two-Sided Properties of Elements in Exchange Rings

  • Dinesh Khurana
  • T. Y. Lam
  • Pace P. Nielsen
Article

Abstract

For any element a in an exchange ring R, we show that there is an idempotent \(\,e\in aR\cap R\,a\,\) such that \(\,1-e\in (1-a)\,R\cap R\,(1-a)\). A closely related result is that a ring R is an exchange ring if and only if, for every aR, there exists an idempotent eR a such that 1−e∈(1−a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules.

Keywords

Two-sided properties Idempotents Exchange elements Suitable elements Exchange rings Suitable rings Endomorphism rings 

AMS Subject Classifications (2010)

16D40 16D50 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsPanjab UniversityChandigarhIndia
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsBrigham Young UniversityProvoUSA

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