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Algebraic Operation of Fuzzy Ideals of a Ring © 2019 JETIR May 2019, Volume 6, Issue 5 www.jetir.org (ISSN-2349-5162) Algebraic Operation of Fuzzy Ideals of a Ring Dr.D.Sivakumar 1 , Dr.S.Anjal Mose 2 , Mrs.R.Padma Priya 3 1 Professor, Department of Mathematics, Annamalai University,Annamalainagar-608002. 2, 3 Assistant Professor, PG & Research Department of Mathematics, St.Joseph’s College of Arts & Science (Autonomous) , Cuddalore-607001 ABSTRACT Fuzzy Sets and Systems[5], R. Kellil using fuzzy numbers On 2-Absorbing Fuzzy Ideals to prove In this paper we introduce a generalization of a some property[6], R. Kellil introduced an New fuzzy ideal in rings. We have obtain necessary approaches on some fuzzy algebraic and sufficient condition for a fuzzy ideal of a ring structures[7].presently J. Jacas, J. Recasens to be a maximal fuzzy ideal. We have also obtain worked on Fuzzy numbers and equality relations the condition for a fuzzy set P defined on the ring [8]. M.Demirci used fuzzy numbers in Fuzzy of integers Z to be fuzzy prime ideal on Z. functions and their fundamental properties[9], Key words: Fuzzy ideals, Algebraic operation, D.Boixader, J. Jacas, J. Recasens said about Triangular fuzzy number, Ring. Fuzzy equivalence relations and the uses[10] and J. M. Anthony, H. Sherwood formed a new 1. INTRODUCTION definition for Fuzzy group redefined[1]. Here we Fuzzy mathematics is one of the subset of used triangular fuzzy number to used some of mathematics; it has divided into two section uncertain algebraic operation are used in some namely fuzzy logic and fuzzy set theory. Fuzzy algebraic properties like sum and product of mathematics is faster growth in research and commutative, associative , etc. development of real life problems with mathematics. In 1965 L.A. Zadeh introduced 2. PRELIMINARIES fuzzy logic,it is used in many application where In this preliminaries section some of few ever uncertainty is used then fuzzy can be applied important definitions related to this paper and well in that area. In 1971 Zadeh used Similarity known properties related with our given topics are relations and fuzzy orderings[1], D. S. Malik, J. used in the following N. Mordeson deals with Fuzzy maximal, radical, Definition 2.1 Fuzzy set: and primary ideals of a ring[2]. A. Rosenfeld used A fuzzy set is considered by a membership fuzzy numbers in Fuzzy groups [3] and W. J. Liu function mapped into a domain space (i.e) applied the new idea for Fuzzy invariant mapping connecting universe of discourse X to subgroups and fuzzy ideals, Fuzzy Sets and the unit period [0, 1] given by A x,()/ x x X Systems [4]. T. Kuraoka, N. Kuroki worked On A fuzzy quotient rings induced by fuzzy ideals, JETIR1905H82 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 511 © 2019 JETIR May 2019, Volume 6, Issue 5 www.jetir.org (ISSN-2349-5162) .Here : X 0,1 is said the degree of dA()() dA A Here (i )12 0, 0 , 0,0.5 dd membership function of the fuzzy set A. dA () dA () (ii )2 0,3 0 , 0.5,1 Definition 2.2 Normal fuzzy set: dd X is called a normal fuzzy set if a fuzzy set A of With the condition 1 It is expressed as the universe of discourse then if there exits at- , least one xX such that ()x =1 AAAA 1( ), 2 ( ), 3 ( ) ; , 0,1 A Definition 2.3 Height of a fuzzy set: (l2 l 1 ) l 1 , 0,0.5 Hence A (l l ) l , 0.5, 1 The leading membership position obtained by 3 2 3 some element in the fuzzy set and it is given by Definition 2.7 Let (G, ⋆) be a group and e its h( A ) sup ( x ) A uniqueness. A fuzzy subset of G is a fuzzy Definition 2.4 Convex fuzzy set: subgroup of G iff A fuzzy set A x,()/A x x X is said to be convex iff for any x12, x X , the membership 1. (e) = 1, function of A satisfies the condition AAA x1(1 ) x 2 min ( x 1 ), ( x 2 ) , [0,1] 2. (l1 ⋆ l2) ≥ min{ ( l1), ( l2)}; Definition2.5 Triangular Fuzzy number: ∀ l1; l2 ∈ G, i -1 A TFN Ap is a subset of in R with following 3. For all l1 ∈ G; ( l1) = ( l1 ). membership function as follows: It is called normal if in addition xl 1 , l x l ()ll 12 −1 21 ( l1⋆ l2 ⋆ l1 ) = (l2), ∀ l1 ∈ G. lx3 i (),x l23 x l Ap ()ll 32 3. OPERATIONS ON FUZZY SETS 0, otherwise Definition 3.1. A fuzzy function on a set S is a mapping f: S × S × S → [0, 1] such that, for x, y Where l1 l 2 l 3 ∈ S there exists a Z∈ S satisfying f(x, y, z) = 1. Definition 2.6: Arithmetic operation of TFN If f is a fuzzy operation on a set S, then f is based on () cuts 1. Commutative If A i is a TFN, then () cuts is given by p 2. Associative A( ),A ( ) for 0,0.5 14 3. Identity A A23( ),A ( ) for 0.5,1 JETIR1905H82 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 512 © 2019 JETIR May 2019, Volume 6, Issue 5 www.jetir.org (ISSN-2349-5162) Results 3.1. If E has an identity e for f and e) = f(a′, a, e) = f(a, a′′, e) = f(a′′, a, e) = 1 f(x, e, x) = 1, ∀x ∈ S, then this identity is and so f(a, a′, e) = f(a, a′′, e) = 1 which unique. implies the equality a′ = a′′. For the right symmetry the proof is trivial. Proof: Here we take two identity of the set S for f, then 1 = f(e′, e, e′) = f(e, e′, e′) and 1 = f(e, e′, e) = f(e′, e, e) and so f(e′, e, e′) = f(e′, e, e) = 1. 4. FUZZY IDEALS ′ But a unique element x such that f(e , e, x) = In this section we can recall the important ′ 1. Subsequently, e = e . basic definition related to fuzzy ideals Definition 3.2 Let f : S×S×S → [0, 1] be a Definition4.1: Classical definition fuzzy operation on a set S and e be the unique Let (R, + ×) be a ring and 0 its identity for +. identity of S for f . An element a ∈ S is symmetric R A fuzzy subset of R is called a fuzzy ideal of if f(a, a′, e) = f(a′, a, e) for some a′ ∈ S. R if Definition 3.3 Let f : S ×S × S → [0, 1] be a fuzzy operation on a set S and e be the unique 1. (0R) = 1, identity of S for f . An element a′ ∈ S if it 2. (a) = (−a), exists such that 3. (a + b) ≥ min { (a), (b)}, f(a, a′, e) = f(a′, a, e) = 1 ∀a, b ∈ R, 4. (a × b) ≥ max { (a), (b)}, is called a symmetric element of a ∈ E. ∀a, b ∈ R. By altering the last state in the above Definition 3.4 Let f : S×S×S→ [0, 1] be a fuzzy operation on a set S. An element a∈S is definition we get a new definition of a fuzzy left (resp. right) regular or cancellable if for ideal of a ring as follows any elements x, y, z ∈S, the equality f(a, x, z) = f(a, y, z) (resp. f(x, a, z) = f(y, a, z)) implies x = Definition4.2: y. It is regular or cancellable if it is left and New definition right regular. Let (R, +, ×) be a ring and 0R its Results 3.2 If f a fuzzy operation on S has an identity for +, a fuzzy subset of R identity e. Any left (resp. right) regular element has is a right fuzzy ideal of R if atmost one symmetric element. 1. (0R) = 1, Proof. 2. (a) = (−a), Suppose that x is left regular and has two symmetric elements a′ and a′′. Then f(a, a′, 3. (a + b) ≥ min { (a), (b)}, JETIR1905H82 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 513 © 2019 JETIR May 2019, Volume 6, Issue 5 www.jetir.org (ISSN-2349-5162) ∀a, b ∈ R, fuzzy ideal different from the characteristic 4. (a) > 0 =⇒ (a × b) > 0, function of R such that B is a fuzzy ideal of R and A ⊂ B ⊆ χR ⇒ B = χR Instead they approached ∀a, b ∈ R. the notion of fuzzy maximal ideals through fuzzy maximal left [and right] ideals. It is called proper if (1) = 0. 5.1.Theorem. Let A be a fuzzy set on R. Then A Proposition: 1 is a fuzzy maximal ideal of R if, and only if, there exist a maximal ideal M of R and a dual atom α ∈ If is a proper fuzzy ideal of R, then for all θ ∈ L such that A(x) = 1, if x ∈ M, and = α, otherwise [0, 1] the set I = {x ∈ R | (x) ≥ θ} is an ideal of the ring (R, +, ×). In addition, if ∀a, b ∈ R, a × b ∈ Remark. 1 The above theorem says that there is a 1-1 correspondence between fuzzy maximal ideals I implies (a × b) = (a). (b), then I is a prime ideal of the ring R. Let us give an example of R and pairs (M, α) where M is a maximal ideal of a fuzzy prime ideal which is not maximal. of R and α is a dual atom in L, if L = [0,1], then there is no dual atom in L.
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