Purely small compressible modules and purely small retractable modules

Ridha, Haider M.; Al-Mothafar, Nuhad S.

doi:10.47974/JDMSC-2718

Open Access Research Article

Purely small compressible modules and purely small retractable modules

^*Haider M. RidhaCorresponding authorhaidar.reda2503p@sc.uobaghdad.edu.iqDepartment of Mathematics College of Science University of BaghdadBaghdad, IraqView full profile → , Nuhad S. Al-Mothafarnuhad.salim@sc.uobaghdad.edu.iqDepartment of Mathematics College of Science University of BaghdadDepartment of Mathematics College of Science University of Baghdad Baghdad, Baghdad, IraqView full profile →

* Corresponding author · click or hover a name for details

pp. 2153–2160Vol. 29Issue 5May 2026DOI: 10.47974/JDMSC-2718 Crossmark XML

Received:: 01 Jan 2026
Published Online:: 22 May 2026
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2718
Pages:: 2153–2160

Abstract

Let S be a commutative ring with identity and let U be an (left) unitary S-module (for brief S-mod). This paper presents a detail and study the concepts of purely small compressible (pu-s-comp) and purely small retractable (pu-s-retract) as a generalization of compressible S-mod and retractable S-mod, and give some of their properties, characteriztions, and examples. On the other hand, we study the relation between pu-s-comp and pu-s-retract and provide illustrative example to support the obtained result.

Keywords

Purely small retractable S-mod Compressible S-mod Retractable S-mod Pu-S sub-mod Pu-S-comp Pu-hollow S-mod

Subject Classifications

Primary 93A30Secondary 49K15

References

[1] M. M. Ali and D. J. Smith, “Pure submodules of multiplication modules,” Contributions to Algebra and Geometry, vol. 45, no. 1, pp. 61–74 (2004).
[2] M. A. Ahmed, I. A. Dhari, and Z. A. Atiya, “Purely small and purely hollow modules,” Iraqi Journal of Science, vol. 63, no. 12, pp. 5487–5495 (2022).
[3] H. Q. Hamdi and N. S. Al-Mothafar, “Study on-purely small sub-modules,”. Journal of Discrete Mathematical Sciences and Cryptography, vol. 27, no. 5, pp. 1723–1728 (2024).
[4] P. Fluery, “Hollow modules and local endomorphism rings,” pacific Journal of Mathematics, vol. 53, pp. 379-385 (1974).
[5] P. C. Diop and M. L. Dia, “On E-small compressible modules,” Journal of Algebra and Related Topics, vol. 9, no. 2, pp. 69–81 (2021).
[6] L. S. Mahmood and I. H. Muslem. “Small compressible module and small retractable module,” International Journal of Applied Mathematics and Statistical Sinenes, vol. 5, no. 2, pp. 15-28 (2016).
[7] M. M. Hameed and A. A. Elewi, “Z-small critically compressible modules,” Journal of Discrete Mathematical Sciences & Cryptography, vol. 28, no. 4-B, pp. 1309–1316 (2025).
[8] W. A. Ali and N. S. Al. Mothafar, “Small prime and quasi-small prime modules,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 24, no. 7, pp. 1967–1971 (2021).

Views: 40Downloads: 9Citations: 0