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.

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.