Quasi-dual Baer modules

Let R be a ring and let M be an R-module with $$S={\text {End}}_R(M)$$ S = End R ( M ) . The module M is called quasi-dual Baer if for every fully invariant submodule N of M, $$\{\phi \in S \mid Im\phi \subseteq N\} = eS$$ { ϕ ∈ S ∣ I m ϕ ⊆ N } = e S for some idempotent e in S. We show that M is quasi-dual Baer if and only if $$\sum _{\varphi \in \mathfrak {I}} \varphi (M)$$ ∑ φ ∈ I φ ( M ) is a direct summand of M for every left ideal $$\mathfrak {I}$$ I of S. The R-module $$R_R$$ 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.