Residual quotient fuzzy subset in near-rings

In 1965, Zadeh [14] introduced the concept of fuzzy subsets and studied their properties on the parallel lines to set theory. In 1971, Rosenfeld [10] defined the fuzzy subgroup and gave some of its properties. Rosenfeld’s definition of a fuzzy group is a turning point for pure mathematicians. Since then, the study of fuzzy algebraic structure has been pursued in many directions such as groups, rings, modules, vector spaces, and so on. In 1981, Das [2] explained the interrelationship between the fuzzy subgroups and their t-level subsets. Fuzzy subrings and ideals were first introduced by Wang-jin Liu [5] in 1982. Subsequently, Mukherjee and Sen [7], Swamy and Swamy [13], Dixit et al. [3], and Rajesh Kumar [4] applied some basic concepts pertaining to ideals from classical ring theory and developed a theory of fuzzy. The notions of fuzzy subnear-ring and ideal were first introduced by Abou-Zaid [1] in 1991. Wang-jin Liu [6] introduced residual quotient fuzzy subset (λ : μ) for any two fuzzy ideals in rings. In this paper, we introduce residual quotient fuzzy subset (λ : μ) for any two fuzzy subsets, which is different from [6], and we characterize some related results in near-rings.