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.

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.

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.

In 1993, Gary Birkenmeier, Henry

Heatherly and Enoch lee6, considers the interconnections between prime ideals

and type one prime ideals in near-rings. 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

.

Most famous open problems in Ring

Theory, known as Koethe’s conjecture, states the sum of two nil left ideals is

nil. A positive answer to Amitsur’s question would lead to a positive solution

of Koethe’s conjecture. 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.

Gerhard

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

Wendt45. 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 interpolation theorem for zero-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.