The effect of singularity on a type of supplemented modules

Let R be a ring, M a right R-module, and S = EndR(M) the ring of all REndomorphisms of M. We say that M is Endomorphism δ-H-supplemented (briefly, E-δ-Hsupplemented) provided that for every φ ∈ S, there exists a direct summand D of M such that M = Imφ + X if and only if M = D + X for every submodule X of M with M/X singular. In this paper, we prove that a non-δ-cosingular module M is E-δ-H-supplemented if and only if M is dual Rickart. We also show that every direct summand of a weak duo E-δ-H-supplemented module inherits the property.