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,

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.Here  : X 0 ,1 is said the degree of dAdA()() A   Here (i )0,012 ,0,0.5    dd membership function of the fuzzy set A. dA () dA () ()0,0ii ,0.5,12  3    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  123(),(),();,0,1    A

Definition 2.3 Height of a fuzzy set: (),0,0.5lll211    Hence A   (),0.5,lll 1 The leading membership position obtained by  323    some element in the fuzzy set and it is given by Definition 2.7 Let (G, ⋆) be a group and e its hAx()sup()  A uniqueness. A fuzzy subset  of G is a fuzzy

Definition 2.4 Convex fuzzy set: subgroup of G iff

A fuzzy set AxxxX,() / is said to be  A  convex iff for any x12 x, X  , the membership 1. (e) = 1, function of A satisfies the condition

AAA xxxx1212(1)min(),() ,[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 , lxl ()ll 12 −1  21 ( l1⋆ l2 ⋆ l1 ) = (l2), ∀ l1 ∈ G.

 lx3   i (),xlxl 23 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

A14( ),A (  ) for   0,0.5 A  3. Identity   A23( ),A (  ) for   0.5,1

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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)},

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∀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. Hence, in this case, R Example: has no fuzzy maximal ideals.

As an application of the above example, let R be Due to the importance of prime ideals in classical the ring Z/10Z. Then the mapping given by ring theory, fuzzy prime has been given much attention. In this section, we discuss the notion of a 0 1 2 3 4 5 6 7 8 9 fuzzy prime ideals and important properties. We also present a characterization of all fuzzy prime 1 0 0.5 0 0.6 0 0.5 0 0.7 0 ideals of Z (a) is as required in the exceeding example so it is a 5.2. Theorem. A fuzzy set P on Z is a fuzzy fuzzy prime ideal. We can also verify it directly. prime ideal if, and only if, it is given by P(n) = 1, For this, to see that (a.b) ≥ max{ (a), (b)} and if p|n = α, otherwise where p is a prime integer or (a.b) > 0 imply (a) > 0 or (b) > 0, it zero and α < 1 suffices to draw the product table and move each Remark.2 For a fixed prime number p, we get the cell with the value of element found in the prime ideal PZ of Z. Fixing α in [0, 1) we get a consistent cell by the product. If we change the unique fuzzy prime ideal of Z as in the above value of (2) = 0.6 in the example to (6) = theorem. We may allow p to vary over all positive 0.7, the subsequent function is still a fuzzy ideal prime numbers and α to take various values in [0, greater than the preceding one and different from 1). Also, the fuzzy subset P of Z given by P(n) = the constant fuzzy ideal 1. 1, if n = 0 = α, otherwise where α

5. FUZZY MAXIMAL AND MINIMAL 5.3.Theorem. Let and β be two fuzzy ideals of IDEALS a ring R. For a decomposition of an element x as a finite sum, x = ∑ aibi where ai,bi ∈ R, we set s(x) In the crisp case, the concept of maximal ideal is = {min( (ai), β(bi))}. Denote by Sx the set of all central to the applications of commutative ring theory to algebraic geometry. Therefore they form the possible decompositions of x as above. If any an important class of ideals in commutative ring element admits a finite number of theory. we discuss its fuzzy counterpart, viz; decompositions, (x) = max Sx, s(x) is an ideal, fuzzy maximal [left, right] ideals. proved that called in the sequel the product of the fuzzy ideals fuzzy maximal ideal A of R cannot be defined as a and β

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Proof. A = min( (ac), β(bd)) ≥ min(max( (a), (c)), max(β(b), β(d))) ={ min( (c), β(b)) = β(b) if β(b) 1. It is clear that among the different decompositions of 0 we have the decomposition 0 ≤ (c) min( (c), β(b)) = (c) otherwise } = 0.1 So η(0) = min{ (0), β(0)} = 1. B = min( (ac′ ), β(bd′ )) ≥ min(max (a), (c ′ 2. If ∑ aibi is a decomposition of x, then )), max(β(b), β(d ′ ))) = (c ′ ) ∑(−ai)bi and ∑(ai)(−bi) are also two C = min( (a ′ c), β(b ′ d)) ≥ min(max( (a ′ ), decompositions of −x. On the other hand, for any decomposition of −x there correspond two (c)), max(β(b ′ ), β(d))) = (c) decompositions of x. Since (−ai) = (ai) and D = min( (a ′ c ′ ), β(b ′ d ′ )) ≥ min(max( (a ′), β(−ai) = β(ai), η(x) = η(−x). (c ′)), max(β(b ′), β(d ′))) = (c ′ ). 3. For the third axiom, it suffices to prove it for two decompositions x = ab and x = a ′ b ′ of x We conclude the above conditions that max {A, and two similar decompositions of y as y = cd and B, C, D} = (c ′) ≥ (c ′) = max{η(x), η(y)) y = c ′ d ′ Proposition 2. CASE:1 Suppose that (a) ≤ β(b), β(b ′ ) ≤ (a ′ Let and β be two fuzzy ideals of a ), (c) ≤ β(d), (c ′ ) ≤ β(d ′ ), (a) ≥ (a ′ ), commutative ring R. The set (c) ≤ (c ′ ) and (c) ≤ (a) ≤ (c ′ ) then η(x) = (a) and η(y) = (c ′ ). On the other I = {x ∈ R | (y)≤ β(xy) ∀y ∈ R} is an ideal of hand the ring R.

A=min{min( (a), β(b)), min( (c), Proof.

β(d))} = min{ (a), (c)} = (c), 1. For all y ∈ R we have (y) ≤ 1 = β(0) = β(0y).

B=min{min( (a), β(b)), min( (c ′), 2. Let x,x′ ∈ I. Then for all y ∈ R, one has (y) ≤ β(xy) and (y) ≤ β(x′y), β(d ′))} = min{ (a), (c ′ )} = (a), (y)≤min{β(xy),β(x′y)}.

C=min{min( (a ′ ), β(b ′ )), min( (c), But min{β(xy),β(x′y)} ≤ β(xy −x′y) and the last β(d)) = min{β(b ′ ), (c)} = (c) or β(b ′ ), expression is just the value β((x−x′)y) and then x−x′ ∈ I. D=min{min( (a ′ ), β(b ′ )), min( (c ′ ), β(d ′ 3. Let now x ∈ I and a ∈ R. Then for any y ∈ R ))} = min{β(b ′ ), (c ′ )} = β(b ′ ). we have (ay) ≥ max{ (a), (y)} so (y) ≤ As (a) is great than β(b ′ ) and (c) we get (ay) ≤ β(x(ay)) = β((xa)y) so xa ∈ I and the conclusion follows. η(x + y) = (a) ≥ µ(a) = min{η(x), η(y)}. 6. Conclusion CASE:2 Suppose that (a) ≤ β(b), β(b ′ ) ≤ (a ′ ), (c) ≤ β(d), (c ′ ) ≤ β(d ′ ), (a) ≥ (a ′ ), We conclude that our aim is to introduce the new definition and examples in generalization of (c) ≤ (c ′ ) and (a) ≤ (c) then η(x) = (a) classical set into fuzzy ideal ring in algebra. The and η(y)= (c ′ ), so max{η(x), η(y)} = (c ′ ). present paper shows the generalize the On the other hand previous proposition to the fuzzy ideals of

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