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 (c) 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 ‘.’ onby 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 Note that I

is a left ideal of R if I satisfies

(1) and (2), and I is a right ideal

of R if I satisfies (1) and (3).

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)

.

We

give some examples of fuzzy ideals of near-rings.

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

1 M. Akram,

Anti fuzzy Lie ideals of Lie algebras, Quasigroups Related Systems 14 (2006)

123-132.

2 Y. Bingxue, Fuzzy

semi-ideal and generalized fuzzy quotient ring, Iran. J. Fuzzy Syst. 5 (2008)

87-92.

3 L. A. Zadeh, Fuzzy

sets, Information and Control 8 (1965) 338-353.

4 S. A. Zaid,

On fuzzy ideals and fuzzy quotient rings of a ring, Fuzzy Sets and Systems 59

(1993) 205-210.

5 M. Zhou, D.

Xiang and J. Zhan, On anti fuzzy ideals of ¡-rings, Ann. Fuzzy Math. Inform.

1(2) (2011) 197-205.

6 A. Rosenfeld, Fuzzy

groups, J. Math. Anal. Appl. 35 (1971) 512-517.

7 W. Liu, Fuzzy

invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139.

8 T. K. Dutta

and B. K. Biswas, Fuzzy ideals of near-rings, Bull. Calcutta Math. Soc.,

89(1997), 447–456.

9 S.M. Hong,

Y.B. Jun, H.S. Kim, Fuzzy ideals in near-rings, Bulletin of Korean

Mathematical Society,

35(3), (1998), 455–464.

10 B.Davvaz,

Fuzzy ideals of near-rings with interval valued membership functions,

Journal of Sciences,

Islamic republic of Iran, 12(2001), no. 2, 171–175.

11 K. H. Kim andY.B. Jun, Anti fuzzy ideals in near rings,

Iranian Journal of Fuzzy

Systems,

Vol. 2 (2005), 71–80.