1. NoteworthyContribution in the field of proposed work Thenotion of fuzzy set was formulated by Zadeh47 and since then there has been aremarkable growth of fuzzy theory. The notion of fuzzy congruence on group wasintroduced by Kuroki and that on universal algebra was studied by Filep andMaurer and by Murali.
Our definition of fuzzy equivalence differs from that ofKuroki in the definition of fuzzy reflexive relation. Some earlier work onfuzzy congruence of a semiring may be found. In the paper ” On fuzzy congruenceof a near-ring module” by T.K.Dutta, B.K. Biswas18 introduce the notion offuzzy submodule and fuzzy congruence of an R-module (where R is a near-ring)and quotient R-module over a fuzzy submodule of an R-module.
We obtainone-to-one correspondence between the set of fuzzy submodules and the set offuzzy congruence corresponding author of an R-module. Lastly, he study fuzzycongruence of quotient R-module over a fuzzy submodule of a R-module and obtaina correspondence theorem.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.
2. Proposed Methodology during the tenure of the research work. Myresearch is concerned with the study of ring and near-ring theory of the basicalgebraic structure and comparing to the arithmetic operations of fuzzy idealsof near-ring. To generalize the basic concept of ideals of rings to fuzzyideals of near-ring.
This purpose first I collect all related data throughgoogle scholor, science direct and shodhganga (INFLIBNET). The basic concept,definition and related theorem of near ring theory are given by pitz. All relatedresearch journals and books shall be procured from google scholar and sci hub.This theory has begun to be applied in multitudes of scientific areas rangingfrom engineering, cryptography and coding theory. However, the basic knowledgeof the ring theory has been pre-assumed and no attempt shall be made to include the proofs of the known resultsto be used during the course of presentwork. 3. Expected outcomeof the proposed work.
We expect that the overall pictureof the research carried out and the recent advancements and new concepts in thefield shall be surveyed. It is almost hundred years since the beginning ofnear-ring theory. At present near-ring theory is one of the most sophisticatedone in pure Mathematics, which has found numerous applications in various areasviz. interpolation theory, group theory, polynomials and matrices. In recentyears its connection with computer science, dynamical systems, rooted treesetc. have also been dealt with.
Thechief motive of this research is to study the properties of near rings andideals of near-ring and compare to the properties of different types of fuzzyideals of a near-ring. Success of fuzzy logic in a wide range of applicationsinspired much interest in fuzzy logic among Mathematicians, Lotfi. A.
Zadeh whointroduced a theory called ” Fuzzy Set theory”. Prof. Zadeh believed that all real worldproblems could be solved with more efficient methods by using the concept fuzzysets. We expect to generalize and extend these concepts of near ring theoryunder fuzzy sets and its applications.Finallythe 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 near-rings theory with a view to project light on some fuzzyideals of near-rings and its generalizations.