Gerald element. The study of near-rings is motivated by

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.

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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.