Yazd University Algebraic Structures and Their Applications 2382-9761 6 2 2019 11 01 H$^*$-condition on the set of submodules of a module 13 20 1427 10.22034/as.2019.1427 EN Ali Reza Moniri Hamzekolaee Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 0000-0002-2852-7870 Journal Article 2019 04 08 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<br />$D$ of $M$ such that $(N+D)/N$ and $(N+D)/D$ are cosingular. We show that over a right perfect right $GV$-ring,<br />a homomorphic image of a $H^*$ duo module satisfies $H^*$. https://as.yazd.ac.ir/article_1427_994a6c2c0b2bd9f31607c92131216852.pdf