On Structures Of Semirings And Congruences On Semirings

In this dissertation, we give a characterization of ring congruences on a semiring ; besides , we discuss the relation of a sublattice of the direct product of the lattices of congruences on a family of semirings and a sublattice of the lattice of congruences on the strong distributive lattice of those semirings ; finally , we discuss generalized fractional semirings and generalized fractional semimodules on those . The main results are given in follow .In Chapter 1 , we give the introduction and preliminaries .In Chapter 3 , we characterize the congruences on a strong distributive lattice of semirings by the congruences on those semirings and prove that a sub-lattice of the direct product of the lattices of congruences on those semirings is isomorphic to a sublattice of the lattice of congruences on the strong distributive lattice of semirings. Finally, we get a necessary and sufficient condition for a quotient semiring of a strong distributive lattice of semirings to be a strong distributive lattice of quotient semirings . The main results are given in follow .In chapter 4 , we mainly get some properties of the generalized fractional semirings and generalized fractional semimodules on those and give a characterization of universal property of generalized fractional semirings.The main results are given in follow.Theorem 4.5 Let R and A are commutative semirings with identities, g : R A is morphism,S' and T are mutiplative sets of R, S C T, and g(S) is a subset of invertible elements of A; g(A) is a subset of cancellable elements of A, then there is a unique morphism h : SRA satisfying hf = g.