On Projectivity, Flatness And Localization Of Semimodules

In this dissertation we investigate some important classes of semimodules such as projective semimodules and flat semimodules. Hom functor, tensor functor and localizations of semimodules are also studied. The dissertation falls into three parts.In the first section, we study the relation between Hom functor and direct product & coproduct. Let R and S be semirings, for rM_s, rU_i, i ∈I , the isomorphism from HomR(Ц_i∈I U_i, M_s)(Hom_R(M_s, П_i∈I U_i), resp.) to П_i∈I Hom_R (U_i, M_s)(П_i∈I Hom_R(M_s, U_i), resp.) is natural in the variable Ms Moreover, the relation between Hom functor and projective semimodules by using R-congruence are discussed. Specially, we prove that left R-semimodule P is projective if and only if Hom_R(P, -) is a covariant congruence exact functor if and only if, for all congruences p over all semimodules M and homomor-phism α : P → M/ρ, there exist homomorphisms α|- : P→ M such that α = π_ρα|-, where π_ρ:M→M/ρ.In the second section, we introduce a new kind of tensor products, and prove three main results. It is shown that every projective semimodule is flat; Let I be an ideals of R and M a left R-semimodule, then R/I RM ≌ M/IM; Also we show that right R-semimodule M is flat if every finitely generated subsemimodule of M is flat. Furthermore, we show that direct limit is a covariant congruence exact functor from category of direct systems to category of semimodules and that the direct limit of direct system of flat semimodules is flat.In the final section of the dissertation, firstly, we improve the operation of sum in the classical left semiring of fractions of semiring with respect to a left 0re set, such that the operation have further rationality. Secondly, we build the classical semimodules of fractions of a semimodule using a straighforward adaptation of the method used for semirings, some basic properties of the semimodule of fractions of a semimodule are also developed. Finally, some local and integral properties of semimodules are studied. It is shown that a semimodule F over a commutative semiring R is flat R-semimodule if and only if Fp is flat Rp-semimodule for all prime ideals P of R, if and only if, Fp is flat Rp-semimodule for all maximal ideals P of R.