The effect of singularity on a type of supplemented modules

Let R be a ring, M a right R-module, and S = End_R(M) the ring of all R-Endomorphisms of M. We say that M is Endomorphism \delta-H-supplemented (briefly, E-\delta-H-supplemented) provided that for every \phi\in S, there exists a direct summand D of M such that M = Im\phi + 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-\delta-cosingular module M is E-\delta-H-supplemented if and only if M is dual Rickart. We also show that every direct summand of a weak duo E-\delta-H-supplemented module inherits the property.

Keywords:

Dual Rickart module , \delta-small-submodule , E-\delta-H-supplemented modules , E-H-supplemented module , H-supplemented module