Abstract
For any element a in an exchange ring R, we show that there is an idempotent \(\,e\in aR\cap R\,a\,\) such that \(\,1-e\in (1-a)\,R\cap R\,(1-a)\). 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
