H∗ -condition on the set of submodules of a module

In this work, we introduce H∗-condition on the set of submodules of a module. Let M be a module. We say M satisfies H∗ provided that for every submodule N of M, there is a direct summand D of M such that (N+D)/N and (N+D)/D are cosingular. We show that over a right perfect right GV-ring, a homomorphic image of a H∗ duo module satisfies H∗.