On M-coidempotent elements and fully coidempotent modules

Abstract

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.