Abstract ideal in -near-rings.In 2009, in our paper title

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.

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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.

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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,
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