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 $H^*$.

The concepts of free modules, projective modules, injective modules and the likeform an important area in module theory. The notion of free fuzzy modules was introducedby Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameriintroduced the concept of projective and injective L-modules. In this paper we give analternate definition for projective L-modules. We prove that e...