A Study of Fuzzy Derivations on Fuzzy Prime Near-Rings

Abstract

The purpose of this paper is to introduce and study the notion of fuzzy derivations on fuzzy nearrings, in the sense of fuzzy groups defined via fuzzy binary operations by Lee and Yuan [13]. Motivated by the classical theory of derivations on near-rings, we extend this framework to the fuzzy setting and establish basic structural properties of fuzzy derivations on fuzzy near-rings and fuzzy prime near-rings. In particular, we investigate how fuzzy derivations interact with fuzzy semigroup ideals and with the fuzzy multiplicative center, and we obtain conditions under which the image of a fuzzy derivation is contained in the fuzzy multiplicative center. As an application, we show that if every value of a nontrivial fuzzy derivation lies in the fuzzy
multiplicative center, then the underlying fuzzy group of the near-ring is necessarily commutative, providing a fuzzy analogue of a well-known result from the ordinary near-ring theory.