Almost Injective Semimodules

Abstract

In this work, injective semimodule has been generalized to almost -injective semimodule. The aim of this research is to study the basic properties of the concept almost- injective semimodules. The semimodule is called almost -injective semimodule if, for each subsemimodule A of and each homomorphism : A , either there exists a homomorphism such that = . Or there exists a homomorphism : Y such that = , where Y is nonzero direct summand of , and is the projection map. A semimodule is almost injective semimodule if it is almost injective relative to all semimodules. Every injective semimodule is almost injective semimodule, if is almost –injective semimodule and is simple, then is -injective. In addition, some related concepts it have been studied and investigated as well.