The effect of singularity on a type of supplemented modules

Document Type : Research Paper

Authors

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

2 Department of Computer Sciences, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

Abstract

Let $R$ be a ring, $M$ a right $R$-module, and $S = End_R(M)$ the ring of all $R$-Endomorphisms of $M.$ We say that $M$ is Endomorphism $\delta$-$H$-supplemented (briefly, $E$-$\delta$-$H$-supplemented) provided that for every $\phi\in S,$ there exists a direct summand $D$ of $M$ such that $M = Im\phi + X$ if and only if $M = D + X$ for every submodule $X$ of $M$ with $M/X$ singular. In this paper, we prove that a non-$\delta$-cosingular module $M$ is $E$-$\delta$-$H$-supplemented if and only if $M$ is dual Rickart. We also show that every direct summand of a weak duo $E$-$\delta$-$H$-supplemented module inherits the property.

Keywords


[1] G. Lee, S. T. Rizvi and C. S. Roman, Dual Rickart modules, Comm. Algebra, 39 No. 11 (2011) 4036-4058.
[2] M. T. Kosan and D. Keskin Tütüncü, H-supplemented duo modules, J. Algebra Appl., 6 No. 6 (2007) 965-971.
[3] D. Keskin Tütüncü, M. J. Nematollahi and Y. Talebi, On H-Supplemented modules, Algebra Colloq., 18 No. Spec01 (2011) 915-924.
[4] M. T. Ko┼čan, δ-lifting and δ-supplemented modules, Algebra Colloq., 14 No. 1 (2007) 53-60.
[5] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, Vol. 147, Cambridge University Press, 1990.
[6] A. R. Moniri Hamzekolaee, H-supplemented modules and singularity, J. Algebraic Struc. Appl., 7 No. 1 (2020) 49-57.
[7] A. R. Moniri Hamzekolaee, A. Harmanci, Y. Talebi and B. Ungor, A new approach to H-supplemented modules via homomorphisms, Turk. J. Math., 42 (2018) 1941-1955.
[8] A. Ç. Özcan, The torsion theory cogenerated by δ-M-small modules and GCO-modules, Comm. Algebra, 35 (2007) 623-633.
[9] Y. Talebi, M. Hosseinpour and T. C. Quynh, On T -δ-noncosingular modules, Sib. Elect. Math. Rep., 15 (2018) 321-331.
[10] Y. Talebi, R. Tribak and A. R. Moniri Hamzekolaee, On H-cofinitely supplemented modules, Bull. Iranian Math. Soc., 30 No. 2 (2013) 325-346.
[11] Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra, 30 No. 3 (2002) 1449-1460.
[12] Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq., 7 No. 3 (2000) 305-318.