Some Studies On Some Classes Of Semirings

In this paper,Armendariz semirings,*-semirings with*-IFP and k-Noetherian semirings are studied.Some properties and theorems in ring theory are extended to semirings,and some new properties and results are given to enrich the theoretical knowledge of semirings.The content of this article is arranged as follows:In Chapter 1,the research background,main research results,preliminaries and symbols used in the article are introduced.In Chapter 2,based on the concept of Armendariz rings,Armendariz semirings are defined,and some properties of Armendariz rings proposed by Anderson and Camillo are extended to semirings.It is proved that in an Armendariz semiring R,when the product of n unary polynomials is zero,the product of their corresponding coefficients is zero.Secondly,it is proved that R is Armendariz if and only if R[x]is Armendariz.At the same time,the equivalent condition that the product of n multivariate polynomials is zero and the product of their corresponding coefficients is zero is given.It is further proved that when n≥ 2,if R is a reduced semiring,then R[x]/(xn)is Armendariz.In Chapter 3,*-semirings with*-IFP are studied.A*-semiring R has*-IFP if for every a ∈R,the right annihilator of a is a*-ideal of R,which is equivalent that ab=0 implies aRb*=0 for all a,b∈R.Some characterizations and examples of this class of semirings are given.As applications,generalized inverses related to*-semirings with*-IFP are studied.For an additive cancellative and Id-complemented*-semiring R having*-IFP,if a ∈ R is reflexive invertible,some equivalent characterizations of a being EP elements,partial isometries,normal elements and strong EP elements are given.In Chapter 4,firstly,a new analogue of the Hilbert’s basis theorem for semirings is obtained.Next,we prove that if the semiring R is left k-Noetherian,then the Ore extension R[x;y,d]is left k-Noetherian where γ is an automorphism of R.In particular,if d is the zero map,the skew polynomial semiring R[x;γ]is obtained,denoted by S(R).Let N(R)denote the set of nilpotent elements of R and γ be an automorphism of R,we get if R is a 2-primal Noetherian semiring,thenγ(N(R))=N(R),and if R is a symmetric k-Noetherian semiring,then S(N(R))=N(S(R)).