Abstract ideals of near-ring. 1. Introduction The concept

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 AbstractTheaim of this paper is to extend the notion of a fuzzy subnear-ring, fuzzy idealsof a near ring, anti fuzzy ideals of near-ring and to give some properties offuzzy ideals and anti fuzzy ideals of a near-ring. Keywords:Near-ring, Near-subring, Ideals of near-ring, Fuzzy set, Fuzzy subring, Fuzzyideals of near-ring, Anti fuzzy ideals of near-ring.1.

    IntroductionTheconcept of fuzzy set was introduced by Zadeh3 in 1965, utilizing whichRosenfeld6 in 1971 defined fuzzy subgroups. Since then, the different aspectsof 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 anear-ring and to give some properties of fuzzy prime ideals of a near-ring. Lui7has studies fuzzy ideal of a ring and they gave a characterization of a regularring. B.

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Davvaz10 introduce the concept of fuzzy ideals of near rings withinterval valued membership functions in 2001. For a complete lattice , introduceinterval-valued -fuzzy ideal(primeideal) 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 normalfuzzy R-subgroup in a near-rings and investigate some related properties. In2005, Syam Prasad Kuncham and Satyanarayana Bhavanari in our paper title “FuzzyPrime ideal of a Gamma-near-ring” introduce fuzzy prime ideal in -near-rings.

In 2009, inour paper title “On the intuitionistic Q-fuzzy ideals of near-rings” introducethe notion of intuitionistic Q-fuzzification of ideals in a near-ring andinvestigate some related properties. F. A. Azam, A. A. Mamun and F.

Nasrindefine the anti fuzzy ideals of near-ring. In this paper we extend the notionof a fuzzy subnear-ring, to study fuzzy ideals of a near ring and to give some differencebetween the properties of fuzzy ideals and anti fuzzy ideals of a near-ring. 2.     PreliminariesForthe sake of continuity we recall some basic definition.Definition 2.1: A set N togetherwith two binary operations + (called addition) and ?(called multiplication) is called a (right) near-ring if:A1: N isa group (not necessarily abelian) under addition;A2:multiplication is associative (so N isa semigroup under multiplication); andA3:multiplication distributes over addition on the right:for any x, y, z in N, itholds 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, thenwe call N a left near-ring.

Near-rings are generalised rings: addition needsnot be commutative and (more important) only one distributive law ispostulated.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 ‘+’ andmatrix multiplication ‘.’ Is defind by the following:                        Becausewe use the multiplication in Z i.

e. a.b=a.

 So.  It is easily verified M2×2 isa 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 suchthat(1)    is a normal subgroup of (2)   (3)    for any  and any  Note that Iis a left ideal of R if I satisfies(1) and (2), and I is a right idealof 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 leftideal of R if  is a fuzzy subnear-ring of R and satisfies: forall (1)    (2)   ,(3)    or  Definition 3.3:Let R be a near-ring and  be a fuzzy subset of R. is called a fuzzy rightideal of R if  is a fuzzy subnear-ring of R and satisfies: forall (1)(2)(3)(4). Wegive 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                                                                 Thewe can easily see that  isan group and  isan semigroup and satisfies left distributive law.

Hance  isa left near-ring. Define a fuzzy subset  by. Then  isa left fuzzy ideal of R. Example 3.5: Let  be a set with two binary operations asfollows: + 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: Ifa 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. Assumethat  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, Andnow,                                                                                                                                                                                                                               (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 fuzzyleft ideal of R if  is a fuzzy subnear-ring of R and satisfies: forall (1)(2),(3) or Definition4.

2:Let R be a near-ring and  be a fuzzy subset of R. is called a anti fuzzyright ideal of R if  is a fuzzy subnear-ring of R and satisfies: forall (1)(2)(3)(4).Proposition4.3:For every anti fuzzy ideals of R,(1)   (2)   (3)   Proof.

(1)                                                                                 .                                                                                   (2)                                                                                             .For all  Since xis arbitrary, we conclude that              (3) Assume that  for all  Then                                                                                                                                                                                                                                                           So,                                                                                              (1)            Also,                                                                                                                                                                                          }                                                            So,                                                                                                          (2)            From equation (1) and (2)            Hence .                                                                                                                                                  5.

References1 M. Akram,Anti fuzzy Lie ideals of Lie algebras, Quasigroups Related Systems 14 (2006)123-132.2 Y. Bingxue, Fuzzysemi-ideal and generalized fuzzy quotient ring, Iran.

J. Fuzzy Syst. 5 (2008)87-92. 3 L. A.

Zadeh, Fuzzysets, 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, Fuzzygroups, J. Math. Anal. Appl.

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K. Duttaand B. K. Biswas, Fuzzy ideals of near-rings, Bull. Calcutta Math. Soc.,89(1997), 447–456.

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S. Kim, Fuzzy ideals in near-rings, Bulletin of KoreanMathematical 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 FuzzySystems,Vol. 2 (2005), 71–80.                           

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