A Generalized Definition of Fuzzy Subrings

Abstract

In this study, under the condition that $L$ is a completely distributive lattice, a generalized definition of fuzzy subrings is introduced. By means of four kinds of cut sets of fuzzy subset, the equivalent characterization of $L$ -fuzzy subring measures are presented. The properties of $L$ -fuzzy subring measures under these two kinds of product operations are further studied. In addition, an $L$ -fuzzy convexity is directly induced by $L$ -fuzzy subring measure, and it is pointed out that ring homomorphism can be regarded as $L$ -fuzzy convex preserving mapping and $L$ -fuzzy convex-to-convex mapping. Next, we give the definition and related properties of the measure of $L$ -fuzzy quotient ring and give a new characterization of $L$ -fuzzy quotient ring when the measure of $L$ -fuzzy quotient ring is 1.