H$^*$-condition on the set of submodules of a module

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar



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 $H^*$.


[1] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhauser, Basel Boston, Berlin, (2006).
[2] D. Keskin Tutuncu, M. J. Nematollahi and Y. Talebi, On H-Supplemented modules, Algebra Colloq. Vol. 18(Spec. 1) (2011), pp. 915-924.
[3] M. T. Kosan and D. Keskin Tutuncu, H-supplemented duo modules, J. Algebra Appl. Vol. 6 No. 6 (2007), pp. 965-971.
[4] S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge, University Press, (1990).
[5] A. R. Moniri Hamzekolaee, A. Harmanci, Y. Talebi and B. Ungor, A new approach to H-supplemented modules via homomorphisms, Turkish J. Math. Vol. 42 (2018), pp. 1941-1955.
[6] N. Orhan Ertas and R. Tribak, Two generalizations of lifting modules, Int. J. Algebra Vol. 13 (2009), pp. 599-612.
[7] Y. Talebi, A. R. Moniri Hamzekolaee and D. Keskin Tutuncu, H-supplemented modules with respect to a preradical, Algebra Discrete Math. Vol. 12 No. 1 (2011), pp. 116-131.
[8] Y. Talebi, A. R. Moniri Hamzekolaee, M. Hosseinpour, A. Harmanci and B. Ungor, Rings for which every cosingular module is projective, Hacet. J. Math. Stat. 48(4) (2019), 973-984.
[9] Y. Talebi and M. J. Nematollahi, Modules with C-condition, Taiwanese J. Math. Vol. 13 No. 5 (2009), pp. 1451{1456.
[10] Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra Vol. 30 No.3 (2002), pp. 1449-1460.
[11] Y. Talebi, R. Tribak and A. R. Moniri Hamzekolaee, On H-co nitely supplemented modules, Bull. Iranian Math. Soc. Vol. 30 No. 2 (2013), pp. 325-346.
[12] R. Tribak, H-supplemented modules with small radicals, East-West J. Math. Vol. 11 No. 2 (2009), pp. 211-221.
[13] Y. Wang and D. Wu, On H-supplemented modules, Comm. Algebra Vol. 40 No. 10 (2012), pp. 3679-3689.