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Exchange elements in rings, and the equation $ XA-BX=I$


Authors: Dinesh Khurana, T. Y. Lam and Pace P. Nielsen
Journal: Trans. Amer. Math. Soc. 369 (2017), 495-516
MSC (2010): Primary 16E50, 16U99
DOI: https://doi.org/10.1090/tran6652
Published electronically: March 2, 2016
MathSciNet review: 3557782
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Abstract: 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 160014, India
Email: dkhurana@pu.ac.in

T. Y. Lam
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: lam@math.berkeley.edu

Pace P. Nielsen
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: pace@math.byu.edu

DOI: https://doi.org/10.1090/tran6652
Keywords: Idempotents, exchange elements, suitable elements, regular elements, clean elements, linear and quadratic equations, suitable rings
Received by editor(s): October 22, 2014
Received by editor(s) in revised form: January 8, 2015
Published electronically: March 2, 2016
Article copyright: © Copyright 2016 by the authors

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