H-Supplemented Modules and Singularity

Let m be a module over a ring r. we call m,δ-h-supplemented provided for every submodule n of m there is a direct summand d of m such that m=n+x if and only if m=d+x for every submodule x of m with m/x singular. we prove that m is δ-h-supplemented if and only if for every submodule n of m there exists a direct summand d of m such that (n+d)/n≪δm/n and (n+d)/d≪δm/d.