# Cofinitely weak generalized $\delta$-supplemented modules

Document Type : Research Paper

Author

Department of Math. Faculty of Basic Science, Babol Noshirvani University of technology, Babol, Iran

10.29252/as.2020.1620

Abstract

We will study modules whose cofinite submodules have weak generalized-$\delta$-supplements.  We attempt to investigate some properties of cofinitely weak generalized $\delta$-supplemented modules.  We will prove for a module $M$ and a semi-$\delta$-hollow submodule $N$ of $M$ that, $M$ is cofinitely weak generalized $\delta$-supplemented if and only if $\frac{M}{N}$ is cofinitely weak generalized $\delta$-supplemented.  Also we show that any $M$-generated module is cofinitely weak generalized $\delta$-supplemented module, where $M$ is cofinitely weak generalized $\delta$-supplemented.  We obtain some other results about this kind of modules.
We will study modules whose cofinite submodules have weak generalized-$\delta$-supplements.  We attempt to investigate some properties of cofinitely weak generalized $\delta$-supplemented modules.  We will prove for a module $M$ and a semi-$\delta$-hollow submodule $N$ of $M$ that, $M$ is cofinitely weak generalized $\delta$-supplemented if and only if $\frac{M}{N}$ is cofinitely weak generalized $\delta$-supplemented.  Also we show that any $M$-generated module is cofinitely weak generalized $\delta$-supplemented module, where $M$ is cofinitely weak generalized $\delta$-supplemented.  We obtain some other results about this kind of modules.

Keywords

#### References

[1] F. Anderson and K. Fuller , Rings and Categories of Modules, Graduate Texts in Mathematics., vol. 13, Springer-Verlag, New York, (1992).
[2] G. Bilhan and T. Guroglu, w-Coatomic Modules, Cankaya. Uni. J. Science. Eng., Vol. 7 No.1, (2010), pp. 17-24.
[3] J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting Modules, Supplements and Projectivity in Module Theory. Frontiers in Math., Birkhauser, Boston, (2006).
[4] F. Y. Eriylmaz and S. Eren, Totally co nitely weak Rad-supplemented modules. International J. Pure and applied Math., Vol. 80 No. 5, (2012), pp. 683-392.
[5] S. H. Mohamed, and B. J. Muller, Continuous and Discrete modules, Cambridge, UK: Cambridge Univ. Press., (1990).
[6] Y. Talebi and B. Talaee, Generalizations of D11 and D+11 modules, A. E. J. Math., Vol. 2 No. 2, (2009), pp. 285293.
[7] Y. Talebi and B. Talaee, On generalized -supplemented modules, V. Journal. Math., Vol. 37 No. 4, (2009), pp. 515-525.
[8] E. Turkmen and A. Pancer, A generalization of Rad-supplemented modules, Int. J. Pure and Applied Math., Vol. 68 No. 4, (2011), pp. 477-485.
[9] E. Turkmen and A. Pancer, On Radical Supplemented Modules, Int. J. C. Cognition, Vol. 7 No. 1, (2009), pp. 62-64.
[10] R. Wisbauer, Foundations of Modules and Ring Theory, Gordon and Breakch, philadephia, (1991).
[11] Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Alg. Coll. Vol. 7 No. 3, (2000), pp. 305-318.