A present it becomes a sophisticated theory with numerous

A near-ring is
an algebraic structure. 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 possessing fascinating and challenging areas of current
mathematical research.  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 Mathematics. 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.

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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. We wish to generalize the different kinds of
fuzzy ideals in near-ring and there properties. We shall also investigate some
of its properties with example.

 

      

1.   
A
brief Review of the work already done in the field

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. 

x

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