# The fuzzy reflexive relation. Some earlier work

Thenotion of fuzzy set was formulated by Zadeh47 and since then there has been aremarkable growth of fuzzy theory. The concept of fuzzy relation on a set wasdefined by Zadeh and other authors like Rosenfeld37 , Tamura, Yeh and Bangconsidered it further. The notion of fuzzy congruence on group was introducedby Kuroki and that on universal algebra was studied by Filep and Maurer and byMurali. Our definition of fuzzy equivalence differs from that of Kuroki in thedefinition of fuzzy reflexive relation. Some earlier work on fuzzy congruenceof a semiring may be found. In the paper ” On fuzzy congruence of a near-ringmodule” by T.

K.Dutta, B.K. Biswas18 introduce the notion of fuzzy submoduleand fuzzy congruence of an R-module (where R is a near-ring) and quotient R-moduleover a fuzzy submodule of an R-module. We obtain one-to-one correspondencebetween the set of fuzzy submodules and the set of fuzzy congruencecorresponding author of an R-module. Lastly, he study fuzzy congruence ofquotient R-module over a fuzzy submodule of a R-module and obtain acorrespondence theorem.

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SalahAbou-Zaid1 (peper title “On Fuzzy subnear-rings and ideals”1991) introducethe notion of a fuzzy subnear-ring, to study fuzzy ideals of a near-ring and togive some properties of fuzzy prime ideals of a near-ring. Lui30 has studiesfuzzy ideal of a ring and they gave a characterization of a regular ring. B.Davvaz19 introduce the concept of fuzzy ideals of near rings with intervalvalued membership functions in 2001. For a complete lattice ,introduce interval-valued -fuzzyideal(prime ideal) of a near-ring which is an extended notion of fuzzyideal(prime ideal) of a near-ring. In2001, Kyung Ho Kim and Young Bae Jun in our paper title ” Normal fuzzyR-subgroups in near-rings”25 introduce the notion of a normal fuzzyR-subgroup in a near-rings and investigate some related properties.

In 2005,Syam Prasad Kuncham and Satyanarayana Bhavanari in our paper title ” FuzzyPrime ideal of a Gamma-near-ring” introduce fuzzy prime ideal in -near-rings.In2009, O. Ratnabala Devi in our paper title ” On the intuitionistic Q-fuzzyideals of near-rings” introduce the notion of intuitionistic Q-fuzzification ofideals in a near-ring and investigate some related properties.GopiKanta Barthakur and Shibu Basak, using the idea of quasi coincidence of a fuzzypoint with a fuzzy set and introduce the notion of -fuzzyprime bi-ideals and semiprime bi-ideals. Also he investigate some relatedproperties of these fuzzy substructures. O.

Ratnabala Devi in our paper title”On -fuzzyessential ideal of near-ring” attempt is to define fuzzy essential ideal ofnear-ring using notions of belongingness ( )and quasi-coincidence(q) of fuzzypoints of sets and study -fuzzyessential ideals of near-rings. He investigate different characterizations ofsuch ideals in terms of their level ideals.                1.   Proposed Methodology during the tenure of the research work. Myresearch concerns the study of ring and near-ring theory of the basic algebraicstructure and comparing to the arithmetic operations of fuzzy ideals ofnear-ring. Apply to the basic concept of ideals of rings to fuzzy ideals ofnear-ring .

This purpose first I collect all related data through googlescholer, science direct and shodhganga (INFLIBNET). The basic concept,definition and related theorem of near ring theory are given by pitz. Allresearch journal and book collect from google scholar and sci hub.

This theoryhas begun to be applied in multitudes of scientific areas ranging fromengineering, cryptography and coding theory. However, the basic knowledge ofthe ring theory has been preassumed and no attempt is made to include theproofs of the known results presented in this synopsis.  2.   Expected outcomeof the proposed work.             These synopsis give an overallpicture of the research carried out and the recent advancements and newconcepts in the field. There are about half a dozen paper on near-rings apartfrom the conference proceedings. Above all there is a online searchabledatabase and bibliography on near-rings. It is almost hundred years since thebeginning of near-ring theory.

At present near-ring theory is one of the mostsophisticated one in pure Mathematics, which has found numerous applications invarious areas viz. interpolation theory, group theory, polynomials andmatrices. In recent years its connection with computer science, dynamicalsystems, rooted trees etc. have also been dealt with.Themain concern of this research is the study of properties of near rings andideals of near-ring and compare to the properties of different types of fuzzyideals of near-ring. Most of the concept of near ring theory related to fuzzyset. Success of fuzzy logic in a wide range application inspired much interestin fuzzy logic among Mathematicians, Lotfi.

A. Zadeh introduced a theory whoseobjects called ” Fuzzy Sets”.  Prof.Zadeh believed that all real world problems could be solved with more efficientmethods by using the concept fuzzy sets. In this synopsis, explain many paperof near ring theory related to fuzzy sets.

I want to generalize and extendthese concept of near ring theory under fuzzy sets and its applications.Nowthe main aim of our proposed work is to study and generalize different types offuzzy ideals, fuzzy congruences and quotient structures in near-ring. Ourobjective is to study of near-rings theory with a view to project light on somefuzzy ideals of near-rings and its generalization.

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