Article ID: MTJPAM-D-25-00065

Title: Modules and matrix rings with SIEP and SSEP properties


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-25-00065; Volume 7 / Issue 1 / Year 2025, Pages 137-145

Document Type: Research Paper

Author(s): ‪Eren Doğan a‪, Mustafa Alkan b

aYuksekova Vocational School, Hakkari University, Hakkari TR-30000, Turkey

bDepartment of Mathematics, Akdeniz University, Antalya TR-07058, Turkey

Received: 6 March 2025, Accepted: 8 April 2025, Published: 18 October 2025

Corresponding Author: Eren Doğan (Email address: erendogan@hakkari.edu.tr)

Full Text: PDF

Abstract

In this study, we investigate modules satisfying the property that the intersection of two direct summands is essential in a direct summand, referred to as SIEP modules and similarly, we study modules where the sum of two direct summands is essential in a direct summand, known as SSEP modules.

Based on the obtained results, this study enhances the understanding of fundamentality conditions in direct sum decompositions by elucidating the relationships between SIEP and SSEP modules and the previously investigated SIP and SSP modules. The findings presented herein contribute to the broader framework of module theory and ring theory, providing new insights for future research on the structural properties of direct sums and their fundamental components.

Subsequently, we discuss the fundamental properties of SSEP and SIEP modules. Furthermore, we examine the structural properties of direct summands of SSEP (SIEP) modules and explore SSEP (SIEP) matrix rings. Finally we prove that K be a ring with identity 1, m a positive integer and R be the ring Matm(K) of all m × m matrices with entries in K, and the h11 denote the matrix in R with (1, 1) entry 1 and all other entries 0, then the R is a right SIEP (SSEP) ring if and only if the free right K-module Km is an SIEP (SSEP) module.

Keywords: SSEP modules, SIEP modules, SSEP matrix rings, SIEP matrix rings

References:
  1. A. N. Abyzov, T. H. N. Nhan and T. C. Quynh, Modules close to and modules, Lobachevskii J. Math. 38 (1), 16–23, 2017.
  2. M. Alkan and A. Harmanci, On summand sum and summand intersection property of modules, Turkish J. Math. 26 (2), 131–147, 2002.
  3. I. Amin, Y. Ibrahim and M. F. Yousif, D3-modules, Comm. Algebra 42 (2), 578–592, 2014.
  4. F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer, New York, 1974.
  5. D. M. Arnold and J. Hausen, A characterization of modules with the summand intersection property, Comm. Algebra 18 (2), 519–528, 1990.
  6. G. Birkenmeier, F. Karabacak and A. Tercan, When is the property inherited by free modules, Acta Math. Hungar. 112 (1–2), 103–106, 2006.
  7. N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Math. (Volume 313), Longman Scientific Technical, Harlow, New York, 1994.
  8. L. Fuchs, Infinite abelian groups, Pure and Applied Mathematics, Academic Press, New York, London, 1970.
  9. J. L. Garcia, Properties of direct summands of modules, Comm. Algebra 17 (1), 73–92, 1989.
  10. K. Goodearl, Ring theory: Nonsingular rings and modules, Marcel Dekker Inc., New York, 1976.
  11. J. Hausen, Modules with the summand intersection property, Comm. Algebra 17 (1), 135–148, 1989.
  12. I. Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954.
  13. F. Karabacak and A. Tercan, Matrix rings with summand intersection property, Czechoslovak Math. J. 53 (3), 621–626, 2003.
  14. F. Karabacak and A. Tercan, On modules and matrix rings with -extending, Taiwanese J. Math. 11 (4), 1037–1044, 2007.
  15. T. Y. Lam, Lectures on modules and rings, Graduate texts in mathematics, Springer World Publ., 2012.
  16. S. H. Mohammed and B. J. Muller, Continous and discrete modules, In: London Mathematical Society Lecture Note Series (Volume 147), Cambridge Univ. Press, Cambridge, 1990.
  17. M. A. Ozarslan, Semiregular, semiperfect and semipotenet matrix rings realtive to an ideal, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 73 (1), 211–221, 2024.
  18. A. Ç. Özcan and M. Alkan, Semiperfect modules with respect to a preradical, Comm. Algebra 34 (3), 841–856, 2006.
  19. R. Rissner and N. J. Werner, Counting core sets in matrix rings over finite fields, 2024; ArXiv:2405.04106.
  20. Y. Talebi and A. R. M. Hamzekolaee, On a generalization of lifting modules via -modules, Bull. Iranian Math. Soc. 48 (2), 343–349, 2022.
  21. G. V. Wilson, Modules with the summand intersection property, Comm. Algebra 14 (1), 21–38, 1986.
  22. R. Wisbauer, Foundations of module and ring theory, Gordon and Breach, London, 1991.

Cite this article

How to cite this article: E. Doğan and M. Alkan, Modules and matrix rings with SIEP and SSEP properties, Montes Taurus J. Pure Appl. Math. 7 (1), 137-145, 2025; Article ID: MTJPAM-D-25-00065.