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

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