Structures And Properties Of Some Semirings

In this paper,we mainly discuss structures and properties of some semirings . In first section,the concept of pseudo ideal is raised .Then,we discuss the relations of pseudo ideals,kernel of homomorphisms and congruences on a division semiring.And the homomorphic basic theorem,the first and the second isomorphic theorems are proved . In the second section,we structure a kind of semirings,namely semidirect product of semirings,and prove an isomorphic theorem of semidirect products .In the third section,we give the characterizations of the relations of all kinds of regular semirings and introduce the concept of pseudo-inverse and the necessary and sufficient conditions of pseudo - invertible element .In the fourth section,we define an equivalence relation on the cartersian product of commutative semiring and its multiplicative subset . Furthermore,structure a kind of commutative semirings .namely fractional semiring .Then,we prove a universal property of fractional semiring,and we discuss the relations between the ideals of commutative semirings and the ideals of fractional semirings. Main results are following as:Theoreml.5 Let S be a division seniring,A be the set of all pseudo ideals of a division semiring B be the set of all congruence relations on S,then there is an injective and surjective map from A into B.Theoreml.il Let 5 and S' be division semirings,let be a surjective morphism of semirings,and let AT be a pseudo ideal of S,if K mkerf,then there exists a unique surjective morphismTheorem2.7 Let S and S' be commutative semirings , T and T' be commutative semigroups with identity element 0 , letg: S-S' be an isomorphism of semirings ,be an isomorphism of semigroups and satisfyingThenthe function f* defined by f*is anisomorphism of SXa.T into S' X a ' T'.Theorem3.21 Given any element x of a semiring R , then x is pseudo invertible in R if and only if it is strongly x-regular in R.Theorem4.13 Let g : R-B be a morphism of semirings satisfying .Then there exists a unique morphism of semirings h: S'R-5 satisfying g=h.Theorem4.15 Let f: R1-R2 be a monomorphism , H1 and H2 be subsemirings and , respectively. Let S, and & be multiplicative sets with respect to H, and H2 , respectively , if/(H1) =H2 and f(S1)=S2, then there exists a monomophism f*: Theorem4.29 Let R be a commutative semiring , let S be the multiplicative set of R , let Q be P-primary ideal of R, then :(1)SP=implies S'Q=S-1R,(2)SP=implies (S'Q) C=Q :(3) the primary ideals of R and the primary ideals of S-1R have one- to- onecorrspondence relation.