Group Of Rough Prime Ideals And Rough Fuzzy Prime Ideal

Connected Rough structure with Algebra structure, Topology structure and Order structure, many new prosperous mathematics branches will appear currently,there have been some articles on connecting Rough structure with Algebra structure. Rough ideals in a semigroup have been studied and the concepts of Rough subsemigroups, Rough ideals been first introduced by Kuroki N. Under the condition of the congruence relation, a rough set of a subsemigroup was proved to be its subsemigroup, while that of a left (right,bisides)ideal was also proved to be its left (right,bisides)ideal. Next, rough sets in a group have been studied and the concepts of Rough subgroups, Rough normal subgroups been first introduced. Under the condition of the congruence relation determined by a given normal subgroup in a group, rough sets of a subgroup was proved to be its subgroup, while that of a normal one was also proved to be its normal subgroup.Homomorphism issues on rough sets and properties of rough sets in rings are discussed by Jinling Zhang. Next,the concept of Fuzzy rough subsemigroups and Fuzzy rough ideals in semigroups are introduced. Then, in rings,the corresponding results of semigroups are generalized.There have six chapters in this paper. The first chapter is Introduction, the theory of Rough set and Fuzzy set are introduced simply.The second chapter is Preparation, the knowledge about rough set that we must know before read this paper is given. The third chapter is Rough prime ideals and Fuzzy rough prime ideals in semigroups,the concept of Rough prime ideals and Fuzzy rough prime ideals in semigroups are introduced. Under the condition of the complete congruence relation, a rough set of a prime ideal in semigroups is proved to be its prime ideal. In the sense of cutsets,the relation between the rough set of a cutset and the cutset of a rough set, a fuzzy rough set of a Fuzzy prime ideal in a semigroup is proved to be its Fuzzy prime ideal, while that of a fuzzy rough set is a Fuzzy prime ideal, iff the rough set of its cutset is a prime ideal. And the homomorphism of them are discussed. The fourth chapter is the properties ofrough set decided by ideals congruence in rings,the properties which is differ from rough sets decided by equivalence congruence relation in rings are discussed. The fifth chapter is Rough prime ideals and Fuzzy rough prime ideals in rings , the corresponding results of a semigroup are generalized in a ring. The last chapter is Rough maximal ideals and Fuzzy rough maximal ideals, the concept of Rough maximal ideals and Fuzzy rough maxiamal ideals are given , the properties of them are discussed.