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. 

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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.