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.

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