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 ≪δ M/N and (N + D)/D ≪δ M/D.
