Abstract
We give a criterion for when idempotents of a ring R which commute modulo the Jacobson radical J(R) can be lifted to commuting idempotents of R. If such lifting is possible, we give extra information about the lifts. A “half-commuting” analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange property. In particular, we show that if R/J(R) is an internal exchange ring and idempotents lift modulo J(R), then R is an internal exchange ring. We also clarify some interesting results in the literature by investigating, and ultimately characterizing, the relationships between the finite (internal) exchange property, the (C3) property, and generalizations of square-free modules. We provide multiple examples delimiting these connections.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: 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.