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

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.