caa a22 4500 606160973 CHVBK 20210128100632.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s10468-015-9524-0 doi (NATIONALLICENCE)springer-10.1007/s10468-015-9524-0 Two-Sided Properties of Elements in Exchange Rings [Elektronische Daten] [Dinesh Khurana, T. Lam, Pace Nielsen] For any element a in an exchange ring R, we show that there is an idempotent e ∈ aR ∩ R a $\,e\in aR\cap R\,a\,$ such that 1 − e ∈ ( 1 − a ) R ∩ R ( 1 − a ) $\,1-e\in (1-a)\,R\cap R\,(1-a)$ . 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∈R a 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. Springer Science+Business Media Dordrecht, 2015 Two-sided properties nationallicence Idempotents nationallicence Exchange elements nationallicence Suitable elements nationallicence Exchange rings nationallicence Suitable rings nationallicence Endomorphism rings nationallicence Khurana Dinesh Department of Mathematics, Panjab University, 160014, Chandigarh, India aut Lam T. Department of Mathematics, University of California, 94720, Berkeley, CA, USA aut Nielsen Pace Department of Mathematics, Brigham Young University, 84602, Provo, UT, USA aut Algebras and Representation Theory Springer Netherlands 18/4(2015-08-01), 931-940 1386-923X 18:4<931 2015 18 10468 https://doi.org/10.1007/s10468-015-9524-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10468-015-9524-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Khurana Dinesh Department of Mathematics, Panjab University, 160014, Chandigarh, India aut NATIONALLICENCE 700 1- Lam T. Department of Mathematics, University of California, 94720, Berkeley, CA, USA aut NATIONALLICENCE 700 1- Nielsen Pace Department of Mathematics, Brigham Young University, 84602, Provo, UT, USA aut NATIONALLICENCE 773 0- Algebras and Representation Theory Springer Netherlands 18/4(2015-08-01), 931-940 1386-923X 18:4<931 2015 18 10468