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.