Quasi-dual Baer modules

Abstract

Let R be a ring and let M be an R-module with S={text {End}}_R(M). The module M is called quasi-dual Baer if for every fully invariant submodule N of M, {phi in S mid Imphi subseteq N} = eS for some idempotent e in S. We show that M is quasi-dual Baer if and only if sum _{varphi in mathfrak {I}} varphi (M) is a direct summand of M for every left ideal mathfrak {I} of S. The R-module R_R is quasi-dual Baer if and only if R is a finite product of simple rings. Other characterizations of quasi-dual Baer modules are obtained. Examples which delineate the concepts and results are provided.