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

Khurana, Dinesh; Lam, T. Y.; Nielsen, Pace

doi:10.1090/tran6652

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.

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