Abstract

The

aim of this paper is to extend the notion of a fuzzy subnear-ring, fuzzy ideals

of a near ring, anti fuzzy ideals of near-ring and to give some properties of

fuzzy ideals and anti fuzzy ideals of a near-ring.

Keywords:

Near-ring, Near-subring, Ideals of near-ring, Fuzzy set, Fuzzy subring, Fuzzy

ideals of near-ring, Anti fuzzy ideals of near-ring.

1.

Introduction

The

concept of fuzzy set was introduced by Zadeh3 in 1965, utilizing which

Rosenfeld6 in 1971 defined fuzzy subgroups. Since then, the different aspects

of algebraic systems in fuzzy settings had been studied by several authors.

Salah Abou-Zaid 4(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. Lui7

has studies fuzzy ideal of a ring and they gave a characterization of a regular

ring. B. Davvaz10 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 Jun11 in our paper title ” Normal fuzzy

R-subgroups in near-rings” 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, 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. F. A. Azam, A. A. Mamun and F. Nasrin define the anti fuzzy

ideals of near-ring. In this paper we extend the notion of a fuzzy

subnear-ring, to study fuzzy ideals of a near ring and to give some difference

between the properties of fuzzy ideals and anti fuzzy ideals of a near-ring.

2. Preliminaries

For

the sake of continuity we recall some basic definition.

Definition 2.1: A set N together

with two binary operations + (called addition) and ?

(called multiplication) is called a (right) near-ring if:

A1: N is

a group (not necessarily abelian) under addition;

A2:

multiplication is associative (so N is

a semigroup under multiplication); and

A3:

multiplication distributes over addition on the right:

for any x, y, z in N, it

holds that (x + y)?z =

(x?z) + (y?z).

This near-ring will be termed as right near-ring.

If n1 (n2 + n3 ) = n1 . n2 + n1 . n3 .

instead of condition A3 the set N satisfies, then

we call N a left near-ring. Near-rings are generalised rings: addition needs

not be commutative and (more important) only one distributive law is

postulated.

Examples 2.2:

(1)

Z be the Set of positive and negative integers with 0. (Z,+) is a group .

Define ‘.’ on Z by a.b=a for all a, b

Z. Clearly (Z,+,.) is a near ring. (2) Let

={ 0,1,2,…,11}. (

,+) is a group under ‘+’ modulo 12.

Define ‘.’ on

by a.b=a for all a ?

. Clearly (

, +, .) is a near ring. (3) Let M2×2={(aij)/ Z

: Z is treated as a near ring}. M2×2 under the operation of ‘+’ and

matrix multiplication ‘.’ Is defind by the following:

Because

we use the multiplication in Z i.e. a.b=a.

So

.

It is easily verified M2×2 is

a near ring.

We denote

instead of

.

Note that

and

but in general

for some

. An ideal I of a near-ring R is a

subset of R such that

(1)

is a normal subgroup of

(2)

(3)

for any

and any

3. Fuzzy ideals of near-rings

Definition 3.1:

Let R be a near-ring and

be a fuzzy subset of R. We say a fuzzy subnear-ring of R if

(1)

(2)

for all

Definition 3.2:

Let R be a near-ring and

be a fuzzy subset of R.

is called a fuzzy left ideal of R if

is a fuzzy subnear-ring of R and satisfies: for

all

(1)

(2)

,

(3)

or

Definition 3.3:

Let R be a near-ring and

be a fuzzy subset of R.

is called a fuzzy right ideal of R if

is a fuzzy subnear-ring of R and satisfies: for

all

(1)

(2)

(3)

(4)

.

Example 3.4:

Let

be a set with two binary operations as follows:

+

a

b

c

d

a

a

b

c

d

b

b

a

d

c

c

c

d

b

a

d

d

c

a

b

.

a

b

c

d

a

a

a

a

a

b

a

a

a

a

c

a

a

a

a

d

a

a

b

b

The

we can easily see that

is an group and

is an semigroup and satisfies left

distributive law. Hance

is a left near-ring. Define a fuzzy subset

by

.

Then

is a left fuzzy ideal of R.

Example 3.5: Let

be a set with two binary operations as

follows:

+

a

b

c

d

a

a

b

c

d

b

b

a

d

c

c

c

d

b

a

d

d

c

a

b

.

a

b

c

d

a

a

a

a

a

b

a

a

a

a

c

a

a

a

a

d

a

b

c

b

Then we can easily see that

is a left near-ring. Define a fuzzy subset

by

. Then

is

a fuzzy left ideal of R, but not

fuzzy right ideal of R, Since

Proposition 3.6: If

a fuzzy subset

of

satisfies the properties

then

(1)

(2)

, for all

Proof.(1)

We have that for any

Hence

.

(2) By

(1), we have that

Hence

Proposition 3.7:

Let

be

a fuzzy ideal of R. If

then

Proof. Assume

that

for all

Then

So,

(1)

Also,

}

So,

(2)

From equation (1) and (2)

Hence

.

Proposition 3.8:

If

is

a fuzzy ideals of near-ring R with

multiplicative identity

. Then

Proof: We know that,

And

now,

(1)

Also

(2)

From equation (1) and (2),

4. Anti fuzzy ideals of near-ring

Definition 4.1:

Let R be a near-ring and

be a fuzzy subset of R.

is called a anti fuzzy left ideal of R

if

is a fuzzy subnear-ring of R and satisfies: for

all

(1)

(2)

,

(3)

or

Definition4.2:

Let R be a near-ring and

be a fuzzy subset of R.

is called a anti fuzzy right ideal of R

if

is a fuzzy subnear-ring of R and satisfies: for

all

(1)

(2)

(3)

(4)

.

Proposition

4.3:

For every anti fuzzy ideals

of R,

(1)

(2)

(3)

Proof.(1)

.

(2)

.

For all

Since x

is arbitrary, we conclude that

(3) Assume that

for all

Then

So,

(1)

Also,

}

So,

(2)

From equation (1) and (2)

Hence

.

5.

References

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