Uj-endomorphism rings

Abstract

In this paper, we introduce and study UJ-modules, that is modules M for which their endomorphism rings E<inf>M</inf> are right UJ. We show, in particular, that: (1) if M is a left UJ-module over a ring R, then M is Dedekind finite; (2) M is a UJ-module iff all clean elements of E<inf>M</inf> are J-clean; (3) M is a clean UJ-module iff E<inf>M</inf> /J(E<inf>M</inf> ) is a Boolean ring and the idempotents lift modulo J(E<inf>M</inf> ) (equivalently, M is a J-clean module); and (4) M is a clean UJ-module such that J(E<inf>M</inf> ) is nil iff M is a conjugate nil clean UJ-module. We also give characterizations of the trivial extension and the (trivial) Morita context, R[x]/(x2) and the tail rings which are right UJ. © 2018 Elsevier B.V., All rights reserved.