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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Exchange elements in rings, and the equation $XA-BX=I$
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by Dinesh Khurana, T. Y. Lam and Pace P. Nielsen PDF
Trans. Amer. Math. Soc. 369 (2017), 495-516


The equation $XA-BX=I$ has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where $B^2=B$, the study of $XA-BX=I$ in a noncommutative ring $R$ leads to several new ways to view and to work with the exchange (or “suitable”) elements in $R$ in the sense of Nicholson. For any exchange element $A\in R$, we show that the set of idempotents $E\in R$ such that $E\in R A$ and $I-E\in R (I-A)$ is naturally parametrized by the roots of a certain left-right symmetric “exchange polynomial” associated with $A$. From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in rings can also be better understood.
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Additional Information
  • Dinesh Khurana
  • Affiliation: Department of Mathematics, Panjab University, Chandigarh 160 014, India
  • MR Author ID: 658568
  • Email:
  • T. Y. Lam
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
  • MR Author ID: 109495
  • Email:
  • Pace P. Nielsen
  • Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
  • MR Author ID: 709329
  • Email:
  • Received by editor(s): October 22, 2014
  • Received by editor(s) in revised form: January 8, 2015
  • Published electronically: March 2, 2016
  • © Copyright 2016 by the authors
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 495-516
  • MSC (2010): Primary 16E50, 16U99
  • DOI:
  • MathSciNet review: 3557782