$H$-supplemented modules and singularity

Document Type : Research Paper


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



Let $M$ be a module over a ring $R$. We call $M$,$\delta$-$H$-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 $\delta$-$H$-supplemented if and only if for every submodule $N$ of $M$ there exists a direct summand $D$ of $M$ such that $(N+D)/N\ll_{\delta} M/N$ and $(N+D)/D\ll_{\delta} M/D$.


[1] M. Hosseinpour, B. Ungor, Y. Talebi and A. Harmanci, A generalization of the class of principally lifting modules, Rocky Mount. J. Math. Vol. 47 No. 5 (2017), pp. 1539–1563.
[2] H. Inankil, S. Halicioglu and Abdullah Harmanci, A generalization of supplemented modules, Algebra Discr. Math. Vol. 11 (2011), pp. 59–74.
[3] S. H. Mohamed and B. J. Mu¨ller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge, University Press, (1990).
[4] A. R. Moniri Hamzekolaee, A. Harmanci, Y. Talebi and B. Ungor, A new approach to H-supplemented modules via homomorphisms, Turk. J. Math. Vol. 42 (2018), pp. 1941–1955.
[5] A. R. Moniri Hamzekolaee, H∗-condition on the set of submodules of a module, J. Algebraic Struc. Appl.Vol. 6 No. 2 (2019), pp. 13–20.
[6] D. Keskin Tu¨tu¨ncu¨, On lifting modules, Comm. Algebra Vol. 28 No. 7 (2000), pp. 3427–3440.
[7] D. Keskin Tutuncu and W. Xue, Generalizations of lifting modules, Acta. Math. Hung. Vol. 91 (2001), pp. 253–261.
[8] D. Keskin Tu¨tu¨ncu¨, M. J. Nematollahi and Y. Talebi, On H-Supplemented modules, Algebra Colloq. Vol. 18 Spec. No. 1 (2011), pp. 915–924.
[9] M. T. Kosan and D. Keskin Tu¨tu¨ncu¨, H-supplemented duo modules, J. Algebra Appl. Vol. 6 No. 6 (2007),pp. 965–971.
[10] M. T. Ko¸san, δ-lifting and δ-supplemented modules, Algebra Colloq. Vol. 14 No. 1 (2007), pp. 53–60.
[11] A. C. Ozcan, The torsion theory cogenerated by δ-M-small modules and GCO-modules, Comm. AlgebraVol. 35 (2007), pp. 623–633.
[12] 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. [13] Y. Talebi and M. J. Nematollahi, Modules with C∗-condition, Taiwanese J. Math. Vol. 13 No. 5 (2009), pp. 1451–1456.
[14] Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra Vol. 30 No. 3 (2002), pp. 1449–1460.
[15] Y. Talebi, R. Tribak and A. R. Moniri Hamzekolaee, On H-cofinitely supplemented modules, Bull. Iranian Math. Soc. Vol. 30 No. 2 (2013), pp. 325–346.
[16] R. Tribak, H-supplemented modules with small radicals, East-West J. Math. Vol. 11 No. 2 (2009), pp. 211-221.
[17] R. Tribak, Y. Talebi, A. R. Moniri Hamzekolaee and S. Asgari, -supplemented modules relative to an ideal, Hacettepe J. Math. Stat. Vol. 45 No. 1 (2016), pp. 107–120.
[18] B. Ungor, S. Halicioglu and A. Harmanci, On a class of supplemented modules, Bull. Malaysian Math. Sci. Soc. Vol. 37 No. 3 (2014), pp. 703–717.
[19] Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. Vol. 7 No. 3 (2000), pp. 305–318.