Kumbhojkar H.V and Bapat M.S[ 2 ]defined not-so- fuzzy fuzzy ideals,.
Palaniappan.N and Arjunan. K[6] definedThe homomorphism, anti
homomorphism of a ...
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 55, 2717 - 2721
Anti–Homomorphisms in Fuzzy Ideals of Rings A. Sheikabdullah Department of Mathematic Syed Hameedha Arts and Science College Kilakarai-623 806, India asheik
[email protected]
K. Jeyaraman Department of Mathematics Alagappa Govt Arts College Karaikudi-630 003, India
[email protected]
Abstract In this paper, a new concept of anti-homomorphism between two fuzzy rings R and R ׀is defined and many results analogous to homomorphism of rings are established. Mathematics Subject Classification: 08A72 Keywords: Fuzzy set, fuzzy ring, fuzzy ideal, fuzzy maximal ideal, fuzzy prime ideal, fuzzy primary, anti-homomorphism in fuzzy rings.
1. INTRODUCTION After the introdution of fuzzy sets by Zadeh.L.A [ 9 ] , several researchers explored on the generalization of the notion of fuzzy set. Kumbhojkar H.V and Bapat M.S[ 2 ]defined not-so- fuzzy fuzzy ideals, Palaniappan.N and Arjunan. K[6] definedThe homomorphism, anti homomorphism of a fuzzy and an anti fuzzy ideals, Chandrasekhara Rao .K and V.Swaminathan [ 1 ] defined the anti homomorphisms in near rings. We define the concept of anti homomorphisms in fuzzy rings and establish some results.
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2. PRELIMINARIES
2.1 Definition : Let X be a non–empty universal set. A fuzzy subset A of X is a function A : X→[0,1]. 2.2 Definition: A fuzzy set μ of a ring R is called a fuzzy sub ring of R if for all x, y ∈ R, μ( x – y ) ≥ min { μ( x ), μ( y ) } μ( x y ) ≥ min { μ( x ), μ( y ) } 2.3 Definition: A fuzzy set μ of a ring R is called a fuzzy ideal of R if for all x, y ∈ R, μ( x – y ) ≥ min { μ( x ), μ( y ) } μ( x y ) ≥ max { μ( x ), μ( y ) } 2.4 Definition: A fuzzy ideal μ of a ring R is called a fuzzy maximal if Im ( μ ) ={1, α } where α∈[ 0,1) and the ideal { x ∈ R / μ( x ) = 1 }is maximal 2.5 Definition: A fuzzy ideal μ of a ring R is called a fuzzy prime if for any two fuzzy ideals σ and θ of R the condition σ θ ⊂ µ implies that σ ⊂ µ or θ ⊂ µ 2.6 Definition: A fuzzy ideal μ of a ring R is called a fuzzy primary if for any two fuzzy ideals σ and θ of R the conditions σ θ ⊆ √µ and σ ⊄ µ together imply that θ ⊆ √µ 2.7 Definition: Let f : R → R ׀be any function, a fuzzy set μ of R is called f-invariant if f(x) = f(y) implies μ(x) = μ(y), x,y ∈ R, 2.8 Definition: Let R and R ׀be two rings ,A mapping f : R → R ׀is called a fuzzy anti-homomorphism if f ( μ + σ ) = f ( μ ) + f ( σ ) and f ( μ σ ) = f (σ ) f ( μ ) 2.9 Remarks: For a fuzzy maximal ideal μ of a ring R , we have ( i ) μ is fuzzy prime and ( ii ) √µ = μ
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3. SOME PROPOSITION
3.1 Proposition : The anti-homomorphic image of a fuzzy ideal of R is a fuzzy ideal of R׀. 3.2 Proposition : The anti-homomorphic pre-image of a fuzzy ideal of R ׀is a fuzzy ideal of R 3.3 Proposition : Let f : R → R ׀be a surjective anti-homomorphism, Let µ ׀is a fuzzy prime ideal of R ׀, then f -1(µ )׀is a fuzzy prime ideal of G. Proof. Let µ and σ be any two fuzzy ideals of R, such that µ σ ⊂ f -1(µ)׀ This implies that f (µ σ ) ⊂ f f -1(µ = )׀µ׀ ⇒ f ( σ )f (µ ) ⊂ = µ ׀because f is an anti-homomorphism ⇒ f ( σ ) ⊂ µ ׀or f (µ ) ⊂ µ ׀because µ ׀is a fuzzy prime ideal of R׀ ⇒ f -1( f ( σ ) ) ⊂ f -1 (µ ) ׀or f -1 ( f (µ ) ) ⊂ f -1 (µ) ׀ ⇒ σ ⊂ f -1 (µ ) ׀or µ ⊂ f -1 (µ) ׀ ⇒ f -1 (µ ) ׀is a fuzzy prime ideal of R׀ 3.4 Proposition : Let f : R → R ׀be an anti-homomorphism.Let µ be any f-invariant fuzzy prime ideal of R ,then f (µ ) is a fuzzy prime ideal of R ׀.
Proof.
Let σ ׀and θ ׀σ be any two fuzzy ideals of R, such that σ ׀θ ⊂ ׀f( µ) ⇒ f -1 (σ ׀θ ⊂ ) ׀f -1f(µ) = µ ⇒ f -1 ( θ ) ׀f -1 (σ ⊂ ) ׀µ
⇒ either f -1 ( θ⊂ ) ׀µ or f -1 (σ ⊂ ) ׀µ since µ is fuzzy prime ideal ⇒ f f -1 ( θ⊂ ) ׀f (µ) or ff -1 (σ ⊂ ) ׀f (µ) ⇒ θ ⊂ ׀f (µ) or σ ⊂ ׀f (µ) ⇒ f (µ) is a fuzzy prime ideal of R׀
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3.5 Proposition : Let f : R → R ׀be a surjective anti-homomorphism. If µ is an finvariant ideal of R and µ fuzzy primary ideal of R, then f (µ ) is a fuzzy primary ideal of R ׀. Proof:
Let σ ׀and θ ׀σ be any two fuzzy ideals of R ׀. such that θ √ ⊂ ׀f( µ) with σ ⊄ ׀f (µ )
⇒ θ √ ⊂ ׀f(µ) ⇒f -1 (σ ׀θ ⊂ ) ׀f -1 √ f(µ) (σ √ = ) ׀f -1 f(µ) = √ µ with f -1 (σ ) ׀f -1f(µ)=µ This implies that f -1 ( θ ) ׀f -1 (σ √⊂ ) ׀µ and f -1 (σ ) ׀not subset of µ because f is anti-homomorphism ⇒f -1 ( θ √⊂ ) ׀µ ⇒ θ ⊂ ׀f f -1 (θ ⊂ ) ׀f (√µ) = √ f(µ) There fore f (µ) is a fuzzy primary ideal of R׀
References [1]
Chandrasekhara Rao .K and Swaminathan, anti-homomorphism in Near rings. Jr of Inst.of maths and computer sciences(math.ser) vol2,No2 (2006),83-88
[2]
Kumbhojkar H.V and and Bapat M.S ,not-so- fuzzy fuzzy ideals ,fuzzy sets and systems ,37(1991),237-243
[3]
Kumbhojkar H.V and and Bapat M.S ,correspondence theorem for fuzzy ideals , fuzzy sets and systems ,41(1991),213-219
[4]
Kumbhojkar H.V and and Bapat M.S ,on semi prime fuzzy ideals , fuzzy sets and systems ,60(1993),219-223
[5]
Olson .D.M, on the homomorphism for hemi rings ,IJMMS,1(1978),439-445
[6]
Palaniappan. N & Arjunan. K , The homomorphism, anti homomorphism of a fuzzy and an anti fuzzy ideals, Varahmihir Journal of Mathematical Sciences, Vo.6 No.1 (2006), 181-188 .
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[7]
Rajesh kumar, Fuzzy Algebra, Delhi University Publication
[8]
Vasantha kandasamy.W.B, Smarandache fuzzy algebra ,American research press, Rehoboth -2003 .
[9]
Zadeh.L.A ,Fuzzy sets , Information and control ,Vol.8, 1965 , 338- 353 .
Received: July, 2010
