Projective and injective semimodules over semirings
In this thesis general notions of projectivity and injectivity for semimodules are defined. This is done by introducing what we call k-projective and i-injective semimodules. The concepts of cogenerator and flatness have also been introduced. In chapter I we give an equivalent definition to projective semimodule. It is shown that the class of all semirnodules such that P is Mk-projective [P is M-injective (P is Mi-injective)] is closed under subsemimodules, factor semimodules and under taking homomorphic images for a k-regular homomorphism. We also characterize the projective, k-projective, injective and i-injective semimodulesi n terms of the Hom functor (chapters I and 111) Also we relate types of injectivity with several types of semi-cogenerators (chapter 111). In chapter 11 we prove that the contravariat functor Hom(-, C) is faithful (semi-faithful) if and only if C is a cogenerator (semi-cogenerator). We introduce the concept of reject for semimodules which plays the important role of radicals in module theory. We show that for any semimodule M and any class μ of semimodules, there is a unique largest factor semimodule of M semi-cogenerated [k-strongly semi-cogenerated (strongly semi-cogenerated)] by μ irrespective of μ semi-cogenerating [k-strongly semi-cogenerating (strongly semi-cogenerating)] M or not. We also characterize semicogenerators in terrns of the Hom functor. Finally, in chapter IV we introduce and investigate flat semimodules and k-flat semimodules. We prove that the semimodule V is flat if and only if the functor (V⊗R-) preserves the exactness of all right regular short exact sequences. We describe the relationship between projectivity and flatness for a certain restricted class of semirings and semimodules. The relationship between flatness and injectivity is also investigated.