A Study On Soft Ring Theory

D.Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool as same as fuzzy sets and rough sets for dealing with various uncertainties arising in economics, engineering, medical science and environmential science. Many researchers applied soft set theory to algebras, information science and computing science, and obtained some important results. We study soft rings based on soft set theory and ring theory in this paper. This paper is composed of four parts. Firstly, concepts of soft rings and idealistic soft rings are introduced by soft sets, some examples are illustrated, and some properties of soft rings are given. In particular, the notions of soft homomorphisms and soft isomorphisms are proposed, and three basic soft isomorphism theorems are established. Secondly, we introduce (∈,∈∨q)- fuzzy subrings, ((∈|￣) , (∈|￣)∨(q|￣))- fuzzy subrings, (α,β)-fuzzy subrings and these kinds of fuzzy ideals, fuzzy bi-ideals and fuzzy quasi-ideals, and give∈-soft set and q - soft set. Then we provide relations between soft rings and these fuzzy subrings. It is point out that there are corresponding results on idealistic soft rings, bi-idealistic soft rings and quasi-idealistic soft rings. Thirdly, regular rings are described by soft set theory, some conditions which are equivalent to regular rings are found. The concept of soft regular rings are introduced, and their properties are discussed. Finally, fuzzy subrings and fuzzy ideals of soft rings are given, some new soft rings are constructed based on fuzzy ideals. Furthermore, the fuzzy isomorphism theorems of soft rings are established.