1.

Introduction

A near-ring (also near-ring or nearring)

is an algebraic structure similar to a ring but

satisfying fewer axioms. The theory of near-rings enjoys the privilege of

not only being deep rooted in many branches of mathematics like geometry, the

theory of automata, non-abelian homological algebra, algebraic topology etc,

but also of prossessing fascinating and challenging areas of current

mathematical research. In fact, the time seems reasonably near for an

historically noteworthy combination of the algebraic theory of near rings with

the fields of nonlinear differential equations, nonlinear functional analysis

and numerical analysis. Twentieth century mathematics has already started

revealing the discipline of mathematics as representing the ultimate in

abstraction, formalization and analytic creativity. The theory of near ring is

a fast growing branch of Abstract Algebra. In 1905, L.E. Dickson17 constructed

the first proper near field by ‘distoring’ the multiplication in a field. These

types of near fields are now called Dickson near-fields.

Two years later, Veblen and

Wedderburn used near-fields to co-ordinate geometric planes. In a monumental

paper, H. Zassenhaus showed in 1936 that all finite near-fields are Dickson

once. Fifty one years later, Zassenhaus showed that there do exit non-Dickson

infinite near-fields of every prime characteristic. For near-rings we now have

a sophisticated structure theory.

Since then the theory of near rings

has been developed much and at present it becomes a sophisticated theory with

numerous application in various areas namely geometries interpolation theory,

group theory, polynomials and matrices. Designs are an important application of

near rings. The use of planner near-rings to get excellent balanced incomplete

designs and experimental designs is probably the best known application of

near-ring to the “outside world”. In recent years its connection with computer

science, automata, dynamical systems, rooted trees, coding theory, cryptography

etc. have also been dealt with. A near-ring is exactly what is needed to

describe the structure of the endomorphisms of various mathematical structures

adequately.

Near-rings are generalisations of rings.

It is natural to generalize various concepts of rings to near-rings. Betch,

Beidleman, Ramakotaiah, Ligh, Clay, Satyanarayana, Chowdhury and others had

generalised various concepts to near-rings. Due to non-ring character of a

near-ring the results have their own beauty. Extensive research work are being

carried out on near-rings and near-ring groups in which structure theory is one

area of importance. Oswald, Beildman, Ligh, Chowdhury and other have done

considerable work on various aspects of near-rings with chain conditions on

annihilators. In 70’s Oswald33 has obtained the structure theory of

near-rings in which each near-ring subgroup is principal. In recent years Pitz34,

Meldrum and others have obtained elegantly the relations between near-rings and

automata, near-rings and dynamical system, semi-near-rings and rooted trees. D.

W. Blackett7 studied simple and semi-simple near-rings around 1953. S.C.

Choudhury, Mason and other have generalised that concept to strictly

semi-simple near-rings. The first ones to use the name “near ring” were

Zassenhaus in 1936 and Blackett and P. Jordan in 1950. Finally, the fifties

brought the start of a rapid development of the theory of near-rings. If in a

ring

we ignore the commutativity of ‘+’ and one of

the distributive laws,

becomes a near ring. If we do not stipulate

the left distributive laws,

is a right near-ring. The set

of all mappings from a group (

to itself with pointwise addition and

composition of mappings serves as a natural example of a near–ring and indeed

all near rings arise as sub near-rings of such near-rings.

The

concept of fuzzy set was introduced by Zadeh47 in 1965, utilizing which

Rosenfeld37 in 1971 defined fuzzy subgroups. Since then, the different

aspects of algebraic systems in fuzzy settings had been studied by several

authors. The notion of fuzzy subnear-ring and fuzzy ideals of near-rings was

introduced by Abou Zaid Salah1. I want to generalize the different kinds of

fuzzy ideals in near-ring and there properties. We also investigate some of its

properties with example.

2.

A

brief Review of the work already done in the field

If

we consider a near-ring N as an N-group over itself, then the existing literature

tells about the dimension theory for ideals (That is., two sided ideals) in

case of commutative near-rings; and one sided ideals in case of associative

(but not commutative) near-rings. So at present we can understand the related

theorems for near-rings in terms of one sided ideals only. Introduce the

necessary concepts and proved the structure theorem for near-rings with respect

to two sided ideals. We studied the concepts: complemented ideal, essential

ideal with respect to two sided ideals of N.

In

1953, D. W. Blackett (Simple and semisimple near ring7) makes an analogous

extension of part of the theory of semisimple ring to semisimple near rings. A

near-ring N is semisimple if it has

no nonzero nilpotent right modules and the right modules satisfy the descending

chain condition. Every nonzero module of a semisimple near ring N contains a nonzero idempotent. Every

minimal nonzero right module M is an irreducible N-space and contains an

idempotent e such that

The important result of these peper is “A

simple near ring is semisimple and has one and only one type of irreducible

space.

In

1954 ,W. E. Deskins (Radical for Near-ring17) restricted to those near-rings

which the descending chain condition for right modules and the requirement that

the zero element of the near ring annihilates the near-ring from the left.

Gerald Berman and Robert J. Silverman introduce

the concept of special class of near ring. Most of the ring properties do not

hold for the endomorphisms of a non-commutative group. In 1961, R. R. Laxton27 introduce the

concept of “Primitive distributively generated near-ring” and ” A Radical and

its theory for distributively generated near-ring. 1965, J.C. Beildeman

introduce in our paper “Quasi-regularity in near-ring(Math. Zeitschr. 89,

224-229)” the concept of quasi-regular R-subgroup and radical subgroup of a near-ring

R with identity are introduced.

Several results asserts that the existence of a

suitably-constrained derivation on a prime near-ring forces the near-ring to be

a ring. A. Boua and L. Oukhtite8 investigate the conditions for a near-ring

to be a commutative ring in our paper title “Derivations on prime near-rings”.

Moreover, examples proving the necessity of the primeness condition are given. The study of

commutativity of 3-prime near-rings by using derivations was initiated by Bell

and Mason3,4 in 1987. Hongan generalizes some results of Bell and Mason3,4

by using an ideal in a 3-prime near-ring instead of the nearing itself. Bell

generalizes several results by using one (two) sided semigroup ideal of the

near-ring in his work. The paper title “Commutativity of near-rings with

derivation by using algebraic substructures” by Ahmed A. M. Kamal and Khalid H.

Al-shaalan26 use subsets and algebraic substructures of a 3-prime nearing R

admitting a derivation d to study the commutativity of the subsets,

the algebraic substructures and the near-ring R under suitable

conditions on d, R and the algebraic substructures.

In

the theory of near-rings the near-rings with identities occupy a role analogous

to that in ring theory of rings with identities. Specially every near-ring may

be embedded in a near-ring with identity. In the peper “The near-rings with

identities on certain finite groups”(Math. Scand.19(1966)) by James R. Clay and

Joseph J. Malone, JR.,15 investigates near-rings with identities, demonstrating

some implications of the existence of the identity element. The study of

near-rings is motivated by consideration of the system generated by the

endomorphisms of a (not necessarily commutative) group. Such endomorphism near-rings

also furnish the motivation for the concept of a distributively generated

(d.g.) near-ring. Although d.g. near-rings have been extensively studied,

little is known about the structure of endomorphism near-rings. In this paper24

results are presented which enable one to give the elements of the endomorphism

near-ring of a given group. Also, some results relating to the right ideal

structure of an endomorphism near-ring are presented (” Near-ring endomorphism”

by J.J. Malone and C.G. Lyons(1969)). It is well-known that a boolean ring29 is

commutative. In this note we show that a distributively generated boolean

near-ring29 is multiplicatively commutative, and therefore a ring. This is accomplished

by using subdirect sum representations of near-rings. The concept ” On Boolean

near-rings” introduce by Steve ligh(1969)29. In the paper “Near-rings in

which each element is a power of itself” by Howard E. Bell2(1970)

generalization of a recent theorem of Ligh on Boolean near-rings.

In

ring theory the notion of quasi-ideal, introduced by O. Steinfeld. It is only

natural to ask whether this notion may be extended to near-rings. The purpose

of this note is to show that this is indeed the case. In 1983, Iwao Yakabe,

paper title ” Quasi-ideals in near-rings”41 introduce the notion of

quasi-ideals in near-rings and consider its elementary properties. Applying

these properties, characterize those near-rings which are near-fields, in terms

of quasi-ideal. A characterization of semi-prime ideals in near-rings by N.J.

Groenewald21. In 1989, Decomposition theorems for periodic rings and

near-rings are proved by Howard E. Bell and Steve Ligh3 on paper ” Some

decomposition theorems for Periodic rings and near-rings”.

Completely

prime ideals have been studied for associative rings by Andrunakievic and

Rjabuhin and also by McCoy. In 1988, N. J. Groenewald21 also define

completely prime radical and show that it coincides with the upper radical

determined by the class of all non-zero near-rings without divisors of zero. He

also give an element wise characterization of this radical.

Howard

E. Bell and Gordon Mason4 study two kinds of derivations in near-rings on

peper ” On Derivations in near-rings and rings”. The first kind, called strong

commutativity-preserving derivation, and second kind called Daif derivations.

An ideal I of a near-ring R is a

type one prime ideal if whenever

,

then

or

. In 1993, Gary

Birkenmeier, Henry Heatherly and Enoch lee6, considers the interconnections between

prime ideals and type one prime ideals in near-rings. The class of all

near-rings for which each prime ideal is type one is investigated and many

examples of such near-rings are exhibited. Various localized distributivity

conditions are found which are useful in establishing when prime ideals will be

type one prime. Jutta hausen and Johnny A. Johnson22 prove that the

Centralizer Near-rings that are rings. Kirby C. Smith generalize that a ring

associated with a near ring. Some recent results on rings deal with

commutativity of prime and semi prime rings admitting suitably-constrained

derivations. It is purpose to extend these results to the case where the

constraints are initially assumed to hold on some proper subset of the

near-ring in the paper title ” On Derivations in near-rings, II” by Howard E.

Bell4 1997.

In

the paper title “Completeness for concrete near-rings” by E. Aichinger, D. Masulovic,

R. Poschel, J. S. Wilson in 2004, a completeness criterion for near-rings over

a finite group is derived using techniques from clone theory. The relationship

between near-rings and clones containing the group operations of the underlying

group shows that the unary parts of such clones correspond precisely to near-rings

containing the identity function.

In

2005, Erhard Aichinger in our paper title ” The near-ring of

congruence-preserving functions on an expanded group” investigate the near-ring

of zero-preserving congruence-preserving

functions on V. Where V be a finite expanded group, e.g. a ring or a group. We

obtain some information on the structure of

from the lattice of ideals of V; for example,

the number of maximal ideals of

is completely determined by the isomorphism

class of the ideal lattice of

.

Kostia

Beidar obtained many nice results in different areas of mathematics.

When he was a

student of Moscow State University he proved that a generalized polynomial

identity on a semiprime ring can be lifted to a maximal right ring of quotients,

a very important result in the theory of generalized identities. Most of

Kostia’s results are connected with ring and nearring theory, but one can also mention

his contribution to Hopf algebras, Jordan and Lie algebras, linear algebra and

mathematical physics. One of the most famous open problems in Ring Theory,

known as Koethe’s conjecture, states that the sum of two nil left ideals (of a

ring) is nil. A positive answer to Amitsur’s question would lead to a positive

solution of Koethe’s conjecture, so Amitsur’s problem was very attractive for

many algebraists. Kostia Beidar published more than 120 research papers and

solved many well-known problems. Our goal is to mention just some of his brilliant

results in ring and nearring theory, and also a brief history of his life.

These concept used by M. A. Chebotar and Y. Font on paper title “The Life

Contribution of Kostla Beidar in ring and near-ring theory” in 2006.

In

2007, C. Selvaraj and R. George39 generalize the concept of strongly prime

near

rings. The paper title ” On Strongly Prime

near

rings” prove some equivalent conditions for strongly prime

near

rings

and radicals

of strongly prime

near

rings

coincides with

where

is strongly prime radicals of left operator

near-ring L of N. The concept of

near

rings, a

generalization of both the concepts near ring and

rings

was introduced by Satyanarayana. Later, several

authors such as Satyanarayana, Booth and Booth, Groenewald studied the ideal

theory of

near

rings. In this paper he prove some equivalent

conditions for strongly prime

near

rings N and radicals

of strongly prime

(equiprime)

near

rings N coincides with the

where

is strongly prime radicals

(equiprime radicals) of left operator near-ring L of N.

Gerhard

Wendt45,46 study the number of zero divisors in zero symmetric near-rings and

are able to give lower and upper bounds for their number for a large class of

near-rings. In case of finite zero symmetric near-rings with identity we can

give our best results and show that the additive group of such a near-ring must

be a p-group if the number of right zero divisors is smaller than a

certain bound. Amongst other results, also the size of ideals of a near-ring

when given its number of zero divisors is discussed on paper title “On Zero

Divisors in Near-rings” in 2009 by Gerhard Wendt.

Gerhard Wendt45

give a short and elementary proof of an interpolation result for primitive

near-rings which are not rings. It then turns out that this re-proves the well

known interpolation theorem for 0-primitive near-rings. Hence, we can offer a

very simple proof for this key result in the structure theory of near-rings in

the paper “A short proof of an interpolation result in near-rings” in 2014. He consider right near-rings, this means the right distributive law

holds, but not necessarily the left distributive law.

In

2012, Ashhan Sezgin, Akin Osman Atagun and Naim Cagman40 in our paper “Soft

intersection near-rings with its applications” define soft intersection near-ring

(soft int near-ring) by using intersection operation of sets.

A. Boua, L. Oukhtite and A. Raji8 in 2015

prove some theorem in the setting of a 3-prime near-ring admitting a suitably

constrained generalized derivation, thereby extending some known results on

derivations. Moreover, we give an example proving that the hypothesis of

3-primeness is necessary. Damir Franetic study loop near-rings, a generalization

of near-rings, where the additive structure is not necessarily associative. And

introduce local loop near-rings and prove a useful detection principle for

localness in our paper title “Local loop near-rings”.

In

2017, many author generalize the concept of near-rings theory.

(1)V.

Chinnadurai and K. Bharathivelan11 in our paper title “Cubic weak Bi-ideals

of near-rings” introduced the new notion of cubic weak bi-ideals of near-rings,

which is the generalized concept of fuzzy weak bi-ideals of near rings. And

also investigated some of its properties with examples.

(2)R.

Jahir Hussain, K. Sampath, and P. Jayaraman24 in our paper title ”

Application of Double- Framed Soft Ideal structures over Gamma near-rings” discuss double-framed soft set theory with

respect to

-near

ring structure. Moreover, investigate double-framed soft mapping with respect

to soft image, soft pre-image and ?-inclusion of

soft sets. Finally, he give some applications of double-framed soft

-near ring to

-near ring theory.

(3)C.

Jaya Subba Reddy and K. Subbarayudu36 proved some results on ” permuting tri-

generalized derivation in prime near rings”.

The concept of a permuting tri- derivation has been introduced Ozturk.

3.

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 concept of fuzzy relation on a set was

defined by Zadeh and other authors like Rosenfeld37 , Tamura, Yeh and Bang

considered it further. 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.

4.

Proposed Methodology during the tenure of the research work.

My

research concerns 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. Apply to the basic concept of ideals of rings to fuzzy ideals of

near-ring . This purpose first I collect all related data through google

scholer, science direct and shodhganga (INFLIBNET). The basic concept,

definition and related theorem of near ring theory are given by pitz. All

research journal and book collect 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 preassumed and no attempt is made to include the

proofs of the known results presented in this synopsis.

5.

Expected outcome

of the proposed work.

Near-rings are one of the

generalized structures of rings. The study and research on near-rings is very

systematic and continuous. These synopsis give an overall picture of the

research carried out and the recent advancements and new concepts in the field.

Conferences devoted solely to near-rings are held once every two years. There

are about half a dozen paper on near-rings apart from the conference

proceedings. Above all there is a online searchable database and bibliography

on near-rings. The two applications of seminear-rings in case of group

automatons and Balanced incomplete block designs (BIBD) and corresponding

application are given in the book “Smarandache near-rings” by W. B. Vasantha

kandasamy. 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

main concern of this synopsis is the study of properties of near rings and

ideals of near-ring and compare to the properties of different types of fuzzy

ideals of near-ring. Success of fuzzy logic in a wide range application

inspired much interest in fuzzy logic among Mathematicians, Lotfi. A. Zadeh ( a

professor in Electrical Engineering and Computer Science at University of

California, Berkeley)( July 1964) introduced a theory whose objects called ”

Fuzzy Sets”. In a narrow sense fuzzy logic refers to a logical system that

generalizes classical two-valued logic for reasoning under uncertainty. Prof.

Zadeh believed that all real world problems could be solved with more efficient

and analytic methods by using the concept fuzzy sets. In this synopsis, explain

many paper of near ring theory related to fuzzy sets. I want to generalize and

extend these concept of near ring theory under fuzzy sets and its applications.

Now

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 of near-rings theory with a view to project light on some

fuzzy ideals of near-rings and its generalization.