TARU PUBLICATIONS
Journal of Discrete Mathematical Sciences and Cryptography cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Pure rationally extending modules

* , ,

* Corresponding author · click or hover a name for details

pp. 1991–1996Vol. 29Issue 5May 2026DOI: 10.47974/JDMSC-2380 Crossmark XML
Received:
01 May 2025
Published Online:
22 May 2026
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2380
Pages:
1991–1996

Abstract

The main purpose for this work, we introduce and study a concept for pure rationally extending modules on the class of rationally extending modules by using pure submodules and direct summands. This paper presents the concept of pure rationally extending modules as a more general concept than rationally extending modules. Throughout this study, many examples and counterexamples are given. In addition, many properties, propositions, and relationships between pure rationally extending modules and some other concepts are discussed.

Keywords

Subject Classifications

Primary 16D5016D60Secondary 16L60

References

[1] K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Pure and Applied Mathematics, no. 33. New York and Basel: Marcel Dekker, Inc. (1976).
[2] M. S. Nayef, Z. M. A. Al-Majeed, Z. R. Shaker, and A. S. Dawood, “Supplement t-extending modules,” J. Discrete Math. Sci. Cryptogr., vol. 27, no. 5, pp. 1591-1598 (2024).
[3] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, vol. 13. New York–Heidelberg: Springer-Verlag (1974).
[4] K. S. Munshid, M. F. Hamid, and J. R. Kider, “Principally pure submodules,” International Journal of Mathematics and Computer Science, vol. 17, pp. 917–922 (2022).
[5] Z. M. A. Al-Majeed and M. S. Nayef, “On supplement rationally-extending modules,” AIP Conference Proceedings, vol. 2398, no. 1, p. 060073 (2022).
[6] M. A. Salih, T. H. Majeed, and M. S. Nayef, “On duality of injective topological module,” Journal of Interdisciplinary Mathematics, vol. 25, no. 5, pp. 1409-1414 (2022).
[7] A. A. Hassan and H. Q. Hamdi, “Generalization of semi-regular modules,” Journal of Interdisciplinary Mathematics, vol. 26, no. 4, pp. 725–73 (2023).
[8] M. S. Abbas and M. F. Hamid, “A Note on Singular and Nonsingular Modules Relative to Torsion Theories,” Mathematical Theory and Modeling, vol. 3, no. 14, pp. 11-15 (2013).
[9] A. T. Hamad and A. A. Elewi, “Z-small submodules and Z-hollow modules,” Iraqi journal of science, vol. 62, no. 8, pp. 2708–2713 (Aug. 2021).
[10] K. S. Munshid, M. F. Hamid, and J. R. Kider, “Principally pure submodules,” International Journal of Mathematics and Computer Science, vol. 17, pp. 917-922 (2022).
[11] D. J. Fieldhouse, “Pure simple and indecomposable rings,” Canadian Mathematical Bulletin, vol. 13, no. 1, pp. 71-78 (1970).
[12] P. F. Smith, “Primary modules over commutative rings,” Glasgow Mathematical Journal, vol. 43, no. 1, pp. 103-111 (2001).
[13] D. J. Fieldhouse, “Pure theories,” Mathematische Annalen, vol. 184, pp. 1-18 (1969).

Views: 40Downloads: 11Citations: 0