Study On(?,?)-Fuzzy Ideals Theory Of Rings

The fuzzy set theory proposed by Zadeh provides a practical mathematical tool for the study of uncertainty.The fuzzy algebra is gradually formed since many scholars apply the fuzzy set theory to groups and rings and successively put forward the concepts of fuzzy semigroup,fuzzy subring as well as various fuzzy ideals after the concept of fuzzy group.With the in-depth study of fuzzy algebra,further development has been gained on the fuzzy algebra but the research on the rings is not complete enough,which need to be further explored.Based on the basic concepts of(?,?)-fuzzy subrings and fuzzy ideals,the definition of intersection,sum and product of(?,?)-fuzzy ideals is introduced,the properties of some operations of(?,?)-fuzzy ideals are further studied and discussed and corresponding conclusions are drew in this paper.Then the concept of(?,?)-fuzzy ideals of(?,?)-fuzzy subrings is defined,the operation of fuzzy ideals of(?,?)-fuzzy subrings is studied in the case of fuzzy ideals operation,making the research content of fuzzy ideals of fuzzy subrings be improved and expanded.Finally,under the(?,?,?)-fuzzy homomorphism,(?,?)-fuzzy subrings and(?,?)-fuzzy ideals are studied.The four tasks are mainly as follows:Firstly,the(?,?)-fuzzy ideals of subrings are introduced and the operations of(?,?)-fuzzy ideals are given.Then some conclusions are obtained by discussing the properties of these operations.Secondly,the concept of(?,?)-fuzzy ideals(bi-ideals,inner ideals)of(?,?)-fuzzy subrings is presented,their related properties are discussed,and some equivalent conditions are obtained as well.Under homomorphic mapping of rings,the corresponding relation between the image and the preimage of(?,?)-fuzzy ideals of(?,?)-fuzzy subrings is established.Thirdly,based on the operations of the(?,?)-fuzzy subrings and(?,?)-fuzzy ideals,further research and discussion are made on the several operation properties of(?,?)-fuzzy ideals of(?,?)-fuzzy subrings.The intersection,sum and product are properly defined,being proved to be closed under algebraic operation and the corresponding conclusions are obtained.Finally,by defining the ?-fuzzy mapping and introducing the(?,?,?)-fuzzy homomorphism of rings,the corresponding relationship between(?,?)-fuzzy subrings and(?,?)-fuzzy ideals under(?,?,?)-fuzzy homomorphism is studied,and the fundamental theorem of(?,?,?)-fuzzy homomorphism of rings is established.In this paper,fuzzy subrings and fuzzy ideals are mainly studied,and the research on fuzzy ideals of fuzzy subrings is tried to be extended.Not only does these research results enrich the fuzzy ring theory and expand the content of fuzzy ideals,but also they lay a good foundation for the further study of(?,?)-fuzzy quotient rings and subrings as well as theorems of homomorphism and isomorphism.