Two-Sided Properties of Elements in Exchange Rings

For any element a in an exchange ring R, we show that there is an idempotent e∈aR∩Ra\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\,e\in aR\cap R\,a\,$\end{document} such that 1−e∈(1−a)R∩R(1−a)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\,1-e\in (1-a)\,R\cap R\,(1-a)$\end{document}. A closely related result is that a ring R is an exchange ring if and only if, for every a∈R, there exists an idempotent e∈Ra such that 1−e∈(1−a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules.