Direct-injective and direct-projective semimodules

Aljebory, Khitam Sahib; Mirza, Rasha Najah

doi:10.47974/JDMSC-2713

Open Access Research Article

Direct-injective and direct-projective semimodules

^*Khitam Sahib AljeboryCorresponding authorkhitamsahib@gmail.comGeneral Directorate of Education Babylon, Ministry of EducationBabylon, IraqView full profile → , Rasha Najah Mirzarasha.mirzah@uokufa.edu.iqDepartment of MathematicsFaculty of Computer Science and MathematicsUniversity of KufaKufa, Iraqhttps://orcid.org/0000-0001-7641-8620View full profile →

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pp. 2117–2122Vol. 29Issue 5May 2026DOI: 10.47974/JDMSC-2713 Crossmark XML

Received:: 01 Jan 2026
Published Online:: 22 May 2026
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2713
Pages:: 2117–2122

Abstract

In this paper, new generalizations of injective (projective) semimodules are introduced. Some of its characterizations are discussed, as well as some properties of quasi-injective and quasi-projective semimodules are extended for direct-injective (projective) semimodule. If a semimodule W is named direct-injective symbolize (Di-injective), if for any direct summand V of W, a monomorphism ∅:V→ W, there is a homomorphism α:W→ W such that α∅ = iv. A semimodule ℘ is said to be direct-projective symbolize (Di-projective), if for any direct summand U of ℘ with projection ν:℘→ U and epimorphism μ:℘→ U, there is an endomorphism α:℘→ ℘ such that μα=ν.

Keywords

Semirings Semimodules Injective (projective) semimodules Quasi-injective (projective) semimodules

Subject Classifications

Primary 16D50Secondary 16L99

References

[1] R. N. Mirza and T. Y. Ghawi, “Radg-lifting Modules,” Journal of Interdisciplinary Mathematics, vol. 26, no. 6, pp. 1001–1013 (2023), doi: 10.47974/JIM-1600.
[2] J. S. Golan, Semirings and Their Applications, Dordrecht, The Netherlands: Springer Science Business Media, pp. 382 (1999), doi:10.1007/978-94-015-9333-5.
[3] T. K. Dutta and M. L. Das, “Singular Radical in Semiring,” Southeast Asian Bulletin of Mathematics, vol. 34, no. 3, pp. 405–416 (2010).
[4] A. J. Naeemah and S. A. A‑Saadi, “The class of semi-modules with direct summands are stable,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 4-B, pp. 1281–1289 (2025), doi: 10.47974/JDMSC-2266.
[5] J. N. Chaudhari and D. R. Bonde, “On subtractive extension of subsemimodules of semimodules,” Journal of the Chungcheong Mathematical Society, vol. 26, no. 1, pp. 37–44 (2013).
[6] K. S. H. Aljebory and A. M. Alhossaini, “Almost Injective Semimodules,” Iraqi J. of Sci, vol. 64, no. 9, pp. 5534–5543 (2023), doi: 10.24996/ijs.2023.64.9.27.
[7] R. R. Nazari and S. Ghalandarzadeh, “Content Semimodules,” Extracta Mathematicae, vol. 32, no. 2, pp. 239–254 (2017).
[8] K. S. H. Aljebory and A. M. A. Alhossaini, “Almost self‑projective semimodules,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 26, no. 7, pp. 1857–1864 (2023), doi: 10.47974/JDMSC‑1675.
[9] A. J. Naeemah and S. Al‑Saadi, “Strongly extending semimodules over a semiring,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 27, no. 5, pp. 1677–1687 (2024), doi: 10.47974/JDMSC‑2009.
[10] Y. Kastove, T. G. Nam, and N. X. Tuyen, “On Subtractive Semisimple Semirings,” Algebra colloquium, vol. 16, no. 3, pp. 415–426 (2009), doi: 10.1142/S1005386709000406.

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