A New generalization of injective semimodule has been presented in this working paper. An Ȑ-semimodule Ӎ is called almost self-injective, if Ӎ is almost Ӎ-injective semimodule. Some properties of this notion have been presented. The relationship of this concept to some concepts has also been clarified as End (Ȑ) of indecomposable almost self-injective semimodules, Rad (Ȑ) also some related notions of it have been studied.
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.
The basis of this paper is to study the concept of almost projective semimodules as a generalization of projective semimodules. Some of its characteristics have been discussed, as well as some results have been generalized from projective semimodules.
In this work, we introduced the Jacobson radical (shortly Rad (Ș)) of the endomorphism semiring Ș = ( ), provided that is principal P.Q.- injective semimodule and some related concepts, we studied some properties and added conditions that we needed. The most prominent result is obtained in section three
-If is a principal self-generator semimodule, then (ȘȘ) = W(Ș).
Subject Classification: 16y60
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