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.

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.