Some Concepts Related to Almost Injective(Projective)Semi modules (JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 2023)
In this research,  the concepts injective and projective semi modules have been generalized to generalized-injective(projective) and essentially-injective semimodules, also the relationship of these concepts with  almost-injective(projective) semimodules have been discussed. The semiring having the properties that every quasi- injective semimodule is injective and every almost self-injective semimodule is injective have been defined. Some results between these concepts  have been obtained like, every almost-injective is generalized-injective semimodule,  the definitions, where a semi module is indecomposable have been discussed.  Dually for almost-projective semimodule. Also obtained that, the direct sum of two quasi-injective Ȑ-semimodule is quasi-injective when  Ȑ is QI-semiring, this fact has been expanded to AQI-semiring.Conclusion: In this paper, several generalization of injective and projective semi modules  have been defined and some concepts with related to almost injective(projective)semimodules have been presented, also  the concepts QI-semiring, AQI-semiring defined, some relationship between these concepts and almost injective(projective) semimodules discussed.