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
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.References
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Additional Information
- Dinesh Khurana
- Affiliation: Department of Mathematics, Panjab University, Chandigarh 160 014, India
- MR Author ID: 658568
- Email: dkhurana@pu.ac.in
- T. Y. Lam
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 109495
- Email: lam@math.berkeley.edu
- Pace P. Nielsen
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 709329
- Email: pace@math.byu.edu
- 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: https://doi.org/10.1090/tran6652
- MathSciNet review: 3557782