The goldie extending property applied to C-closed submodules

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq.

Abstract

In this study, we define a module $\mathcal{H}$ to be Goldie CCLS if and only if there exists a direct summand $\mathcal{B}$ of $\mathcal{H}$ such that $\mathcal{L} \cap \mathcal{B}$ is essential in each of $\mathcal{L}$ and $\mathcal{B}$ for each c-closed submodule $\mathcal{L}$ of $\mathcal{H}$. We examine the structural characteristics of Goldie CCLS modules and identify the connections with the other extending generalizations. We discuss the theory of decomposition. Using examples, we derive several essential features and characterizations of Goldie CCLS modules.

Keywords

References

[1] E. Akalan, G. F. Birkenmeier and A. Tercan, Goldie extending modules, Commun. Algebra, 37 No. 2 (2009) 663-683.
[2] G. F. Birkenmeier, J. B. Muller and S. T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Commun. Algebra, 30 (2002) 1395-1415.
[3] N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics Series 313, Longman, New York, 1994.
[4] A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc., 10 (1960) 201-220.
[5] K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Marcel Dekker, New York, 1976.
[6] I. Kikumasa and Y. Kuratomi, On Goldie extending modules with condition (C2), Commun. Algebra, 44 No. 9 (2016) 4041-4046.
[7] E. M. Kamil, On CCLS modules, AIP Conf. Proc., 2834 (2023) 080128.
[8] E. M. Kamil and S. Al-Aeashi, An approach to generalized extending modules via ec-closed submodules, Iraqi J. Sci., 64 No. 10 (2023) 5158-5164.
[9] E. M. Kamil and B. Davvaz, A generalization of ECS-modules via c-closed submodules, Afr. Math., 36 (2025) 64.
[10] E. M. Kamil and B. Davvaz, A generalization of singular modules in terms of purely extending property, Ann. Univ. Ferrara, 71 (2025) 7.
[11] E. M. Kamil and B. Davvaz, Modules in which fully invariant z-closed submodules are direct summands, Palest. J. Math., 14 No. 2 (2025) 562-568.
[12] P. F. Smith, Modules for which every submodule has a unique closure, In: Ring Theory (Granville, OH, 1992), pp. 302-313, World Scientific, River Edge, NJ, 1993.
[13] P. F. Smith and A. Tercan, Continuous and quasi-continuous modules, Houston J. Math., 18 No. 3 (1992) 339-348.
[14] P. F. Smith and A. Tercan, Generalizations of CS-modules, Commun. Algebra, 21 (1993) 1809-1847.
[15] A. Tercan and C. C. Yücel, Module Theory, Extending Modules and Generalizations, Frontiers in Mathematics, Birkhäuser/Springer, Switzerland, 2016.
[16] A. Tercan, R. Yasar and C. C. Yücel, Goldie extending modules on the class of exact submodules, Commun. Algebra, (2022) 1363-1371.
[17] A. Tercan, R. Yasar and C. C. Yücel, Goldie extending modules on the class of z-closed submodules, Bull. Korean Math. Soc., (2022) 453-468.
[18] C. C. Yücel, On generalized FI-extending modules, Kyungpook Math. J., 60 No. 1 (2020) 44-51.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 22 May 2026

Files

Share

How to cite

Statistics