Abstract
In this study, we define the concept of an ω-fuzzy set ω-fuzzy subring and show that the intersection of two ω-fuzzy subrings is also an ω-fuzzy subring of a given ring. Moreover, we give the notion of an ω-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an ω-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an ω-fuzzy quotient subring. We also define the idea of a support set of an ω-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe ω-fuzzy homomorphism and ω-fuzzy isomorphism. We establish an ω-fuzzy homomorphism between an ω-fuzzy subring of the quotient ring and an ω-fuzzy subring of this ring. We constitute a significant relationship between two ω-fuzzy subrings of quotient rings under the given ω-fuzzy surjective homomorphism and prove some more fundamental theorems of ω-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of ω-fuzzy isomorphism.
Highlights
The number of sets is inherently equipped with two binary operations: addition and multiplication
Fundamental Theorem of ω-Fuzzy Isomorphism of ω-Fuzzy Subrings we introduce the concept of ω-fuzzy homomorphism and ω-fuzzy isomorphism
We investigate the concept of the ω-fuzzy homomorphism relation between any two ω-fuzzy subrings
Summary
The number of sets is inherently equipped with two binary operations: addition and multiplication. The theory of fuzzy sets since has had interesting application areas in both theoretical and practical studies from life science to physical sciences, computer science to health science and engineering to humanities This reason-based thinking helps to master the arts and skills in order to attain certain jobs. The aspiration to form a sketch of this unique technique of an ωfuzzy set in the study of fuzzy ring theory served as the main motivation to propose and develop the theory of ω-fuzzy subrings. Another keynote of this paper is to define the ω-fuzzy homomorphism and prove the numerous fundamental theorems of ω-fuzzy homomorphism analog to classical homomorphism. We discuss three fundamental theorems of ω-fuzzy isomorphism
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