On Studies On Some Classes Of Semirings By Using Of Ideals

Ideal is one of the most important tools in the study of semiring theory. Recently, M. Shabir, A. Ali and Samian Batool discuss the properties of regular semiring by using of one-side ideal, ideal, quasi-ideal and bi-ideal and give the characterization of regular semiring. In this paper, we mainly studyπ-regular semirings,completelyπ-regular semirings and quasi-intra-regular semirings by using of the several typies of ideals.The paper consists of four chapters. In chapter 1, we introduce the history back-groud of quasi-ideal and bi-ideal in semigroup and semiring and some basic definations of different typies of idea in the semiring and some basic properties about quasi-ideal in the semiring. In chapter 2. firstly, we give some fundamental properties about ideals inπ- regular semirings; secondly, we characterizeπ- regular semirings by ideals, quasi-ideals and bi-ideals ,then we extend some corresponding results of paper[2]. At the same time, we simply study regular left duo semiring . In chapter 3, we give characterization of completelyπ- regular semirings and study the Green-L relation and Green-R relation of them. In chapter 4, we study some properties of quasi-intra -regular semirings.