1.   
Noteworthy
Contribution in the field of proposed work

               

The
notion of fuzzy set was formulated by Zadeh47 and since then there has been a
remarkable growth of fuzzy theory. The notion of fuzzy congruence on group was
introduced by Kuroki and that on universal algebra was studied by Filep and
Maurer and by Murali. Our definition of fuzzy equivalence differs from that of
Kuroki in the definition of fuzzy reflexive relation. Some earlier work on
fuzzy congruence of a semiring may be found. In the paper ” On fuzzy congruence
of a near-ring module” by T.K.Dutta, B.K. Biswas18 introduce the notion of
fuzzy 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 obtain
one-to-one correspondence between the set of fuzzy submodules and the set of
fuzzy congruence corresponding author of an R-module. Lastly, he study fuzzy
congruence of quotient R-module over a fuzzy submodule of a R-module and obtain
a correspondence theorem.

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Salah
Abou-Zaid1 (peper title “On Fuzzy subnear-rings and ideals”1991) introduce
the notion of a fuzzy subnear-ring, to study fuzzy ideals of a near-ring and to
give some properties of fuzzy prime ideals of a near-ring. Lui30 has studies
fuzzy 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 interval
valued membership functions in 2001. For a complete lattice

,
introduce interval-valued

-fuzzy
ideal(prime ideal) of a near-ring which is an extended notion of fuzzy
ideal(prime ideal) of a near-ring.

In
2001, Kyung Ho Kim and Young Bae Jun in our paper title ” Normal fuzzy
R-subgroups in near-rings”25 introduce the notion of a normal fuzzy
R-subgroup in a near-rings and investigate some related properties. In 2005,
Syam Prasad Kuncham and Satyanarayana Bhavanari in our paper title ” Fuzzy
Prime ideal of a Gamma-near-ring” introduce fuzzy prime ideal in

-near-rings.

In
2009, O. Ratnabala Devi in our paper title ” On the intuitionistic Q-fuzzy
ideals of near-rings” introduce the notion of intuitionistic Q-fuzzification of
ideals in a near-ring and investigate some related properties.

Gopi
Kanta Barthakur and Shibu Basak, using the idea of quasi coincidence of a fuzzy
point with a fuzzy set and introduce the notion of

-fuzzy
prime bi-ideals and semiprime bi-ideals. Also he investigate some related
properties of these fuzzy substructures. O. Ratnabala Devi in our paper title
“On

-fuzzy
essential ideal of near-ring” attempt is to define fuzzy essential ideal of
near-ring using notions of belongingness (

)
and quasi-coincidence(q) of fuzzy
points of sets and study

-fuzzy
essential ideals of near-rings. He investigate different characterizations of
such ideals in terms of their level ideals.

              

 

2.   
Proposed Methodology during the tenure of the research work.

 

My
research is concerned with the study of ring and near-ring theory of the basic
algebraic structure and comparing to the arithmetic operations of fuzzy ideals
of near-ring. To generalize the basic concept of ideals of rings to fuzzy
ideals of near-ring. This purpose first I collect all related data through
google scholor, science direct and shodhganga (INFLIBNET). The basic concept,
definition and related theorem of near ring theory are given by pitz. All related
research journals and books shall be procured from google scholar and sci hub.
This theory has begun to be applied in multitudes of scientific areas ranging
from engineering, cryptography and coding theory. However, the basic knowledge
of the ring theory has been pre-assumed and no attempt shall be  made to include the proofs of the known results
to be used during the course of  present
work.

 

 

3.   
Expected outcome
of the proposed work.

 

            We expect that the overall picture
of the research carried out and the recent advancements and new concepts in the
field shall be surveyed. It is almost hundred years since the beginning of
near-ring theory. At present near-ring theory is one of the most sophisticated
one in pure Mathematics, which has found numerous applications in various areas
viz. interpolation theory, group theory, polynomials and matrices. In recent
years its connection with computer science, dynamical systems, rooted trees
etc. have also been dealt with.

The
chief motive of this research is to study the properties of near rings and
ideals of near-ring and compare to the properties of different types of fuzzy
ideals of a near-ring. Success of fuzzy logic in a wide range of applications
inspired much interest in fuzzy logic among Mathematicians, Lotfi. A. Zadeh who
introduced a theory called ” Fuzzy Set theory”.  Prof. Zadeh believed that all real world
problems could be solved with more efficient methods by using the concept fuzzy
sets. We expect to generalize and extend these concepts of near ring theory
under fuzzy sets and its applications.

Finally
the main aim of our proposed work is to study and generalize different types of
fuzzy ideals, fuzzy congruences and quotient structures in near-ring. Our
objective is to study near-rings theory with a view to project light on some fuzzy
ideals of near-rings and its generalizations.