Some Concepts Related to Almost Injective(Projective)Semi modules

Abstract

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.