Abstract
The main goal of this paper is to introduce the notion of intuitionistic fuzzy normed rings and to establish basic properties related to it. We extend normed rings by incorporating the idea of intuitionistic fuzzy to normed rings, we develop a new structure of fuzzy rings which will be called an intuitionistic fuzzy normed ring. As an extension of intuitionistic fuzzy normed rings, we define the concept of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals. Some essential operations specially subset, complement, union, intersection and several properties relating to the notion of generalized intuitionistic fuzzy normed rings are identified. Homomorphism and isomorphism of intuitionistic fuzzy normed subrings are characterized. We identify the image and the inverse image of intuitionistic fuzzy normed subrings under ring homomorphism and study their elementary properties. Some properties of intuitionistic fuzzy normed rings and relevant examples are presented.
Highlights
After the inception of the notion of fuzzy set as a generalization of a crisp set by Zadeh [1], which laid to the establishment of the fuzzy set theory
In 1992 [8], Dixit et al introduced the notion of intuitionistic fuzzy groups, followed by a study of the properties of intuitionistic fuzzy subgroups and intuitionistic fuzzy subrings conducted by Hur et al in [9]
We review some basic ideas of intuitionistic fuzzy set, intuitionistic fuzzy subring and intuitionistic fuzzy ideal that are related to the study in this work
Summary
After the inception of the notion of fuzzy set as a generalization of a crisp set by Zadeh [1], which laid to the establishment of the fuzzy set theory. Liu in [3] introduced the definition of fuzzy ring and discussed fuzzy subrings and fuzzy ideals and presented some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, fuzzy ideals and proved some related properties. In 1992 [8], Dixit et al introduced the notion of intuitionistic fuzzy groups, followed by a study of the properties of intuitionistic fuzzy subgroups and intuitionistic fuzzy subrings conducted by Hur et al in [9]. Marashdeh and Salleh generalized Dib’s notion for fuzzy rings based on the notion of fuzzy space in [11] by applying the notion of fuzzy space to intuitionistic fuzzy space and to introduce related concepts to present a new formulation of intuitionistic fuzzy group and intuitionistic fuzzy ring.
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