Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2016, Article ID 7821678, 5 pages http://dx.doi.org/10.1155/2016/7821678
Research Article Radical Structures of Fuzzy Polynomial Ideals in a Ring Hee Sik Kim,1 Chang Bum Kim,2 and Keum Sook So3 1
Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea Department of Mathematics, Kookmin University, Seoul 136-702, Republic of Korea 3 Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea 2
Correspondence should be addressed to Keum Sook So;
[email protected] Received 13 November 2015; Accepted 28 February 2016 Academic Editor: Cengiz C ¸ inar Copyright © 2016 Hee Sik Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study its properties. Given a 𝑛 fuzzy ideal 𝛽 of 𝑅 and a homomorphism 𝑓 : 𝑅 → 𝑅 , we show that if 𝑓𝑥 is the induced homomorphism of 𝑓, that is, 𝑓𝑥 (∑𝑖=0 𝑎𝑖 𝑥𝑖 ) = 𝑛
∑𝑖=0 𝑓(𝑎𝑖 )𝑥𝑖 , then 𝑓𝑥 −1 [(√𝛽)𝑥 ] = (√𝑓−1 (𝛽))𝑥 .
1. Introduction
2. Preliminaries
Zadeh [1] introduced the notion of a fuzzy subset 𝐴 of a set 𝑋 as a function from 𝑋 into [0, 1]. Rosenfeld [2] applied this concept to the theory of groupoids and groups. Liu [3] introduced and studied the notion of the fuzzy ideals of a ring. Following Liu, Mukherjee and Sen [4] defined and examined the fuzzy prime ideals of a ring. The concept of fuzzy ideals was applied to several algebras: 𝐵𝑁-algebras [5], 𝐵𝐿-algebras [6], semirings [7], and semigroups [8]. Ersoy et al. [9] applied the concept of intuitionistic fuzzy soft sets to rings, and Shah et al. [10] discussed intuitionistic fuzzy normal subrings over a nonassociative ring. Prajapati [11] investigated residual of ideals of an 𝐿-ring. Dheena and Mohanraj [12] obtained a condition for a fuzzy small right ideal to be fuzzy small prime right ideal. The present authors [13] introduced the notion of a fuzzy polynomial ideal 𝛼𝑥 of a polynomial ring 𝑅[𝑥] induced by a fuzzy ideal 𝛼 of a ring 𝑅 and obtained an isomorphism theorem of a ring of fuzzy cosets of 𝛼𝑥 . It was shown that a fuzzy ideal 𝛼 of a ring is fuzzy prime if and only if 𝛼𝑥 is a fuzzy prime ideal of 𝑅[𝑥]. Moreover, we showed that if 𝛼𝑥 is a fuzzy maximal ideal of 𝑅[𝑥], then 𝛼 is a fuzzy maximal ideal of 𝑅. In this paper we investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study their properties.
In this section, we review some definitions which will be used in the later section. Throughout this paper unless stated otherwise all rings are commutative rings with identity. Definition 1 (see [3]). A fuzzy ideal of a ring 𝑅 is a function 𝛼 : 𝑅 → [0, 1] satisfying the following axioms: (1) 𝛼(𝑥 + 𝑦) ≥ min{𝛼(𝑥), 𝛼(𝑦)}. (2) 𝛼(𝑥𝑦) ≥ max{𝛼(𝑥), 𝛼(𝑦)}. (3) 𝛼(−𝑥) = 𝛼(𝑥) for any 𝑥, 𝑦 ∈ 𝑅. Definition 2 (see [2]). Let 𝑓 : 𝑅 → 𝑆 be a homomorphism of rings and let 𝛽 be a fuzzy subset of 𝑆. We define a fuzzy subset 𝑓−1 𝛽 of 𝑅 by 𝑓−1 𝛽(𝑥) fl 𝛽(𝑓(𝑥)) for all 𝑥 ∈ 𝑅. Definition 3 (see [2]). Let 𝑓 : 𝑅 → 𝑆 be a homomorphism of rings and let 𝛼 be a fuzzy subset of 𝑅. We define a fuzzy subset 𝑓(𝛼) of 𝑆 by 𝑓 (𝛼) (𝑦) {sup {𝛼 (𝑡) | 𝑡 ∈ 𝑅, 𝑓 (𝑡) = 𝑦} fl { 0 {
if 𝑓−1 (𝑦) ≠ 0, if 𝑓−1 (𝑦) = 0.
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Definition 4 (see [2]). Let 𝑅 and 𝑆 be any sets and let 𝑓 : 𝑅 → 𝑆 be a function. A fuzzy subset 𝛼 of 𝑅 is called an 𝑓-invariant if 𝑓(𝑥) = 𝑓(𝑦) implies 𝛼(𝑥) = 𝛼(𝑦), where 𝑥, 𝑦 ∈ 𝑅. Zadeh [1] defined the following notions. The union of two fuzzy subsets 𝛼 and 𝛽 of a set 𝑆, denoted by 𝛼 ∪ 𝛽, is a fuzzy subset of 𝑆 defined by (𝛼 ∪ 𝛽) (𝑥) fl max {𝛼 (𝑥) , 𝛽 (𝑥)}
(2)
for all 𝑥 ∈ 𝑆. The intersection of 𝛼 and 𝛽, symbolized by 𝛼∩𝛽, is a fuzzy subset of 𝑆, defined by (𝛼 ∩ 𝛽) (𝑥) fl min {𝛼 (𝑥) , 𝛽 (𝑥)}
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for all 𝑥 ∈ 𝑆. Theorem 5 (see [13]). Let 𝛼 : 𝑅 → [0, 1] be a fuzzy ideal of a ring 𝑅 and let 𝑓(𝑥) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥2 + ⋅ ⋅ ⋅ + 𝑎𝑛 𝑥𝑛 be a polynomial in 𝑅[𝑥]. Define a fuzzy set 𝛼𝑥 : 𝑅[𝑥] → [0, 1] by 𝛼𝑥 (𝑓(𝑥)) fl min𝑖 {𝛼(𝑎𝑖 ) | 𝑎𝑖 ’s are coefficients of 𝑓(𝑥)}. Then 𝛼𝑥 is a fuzzy ideal of 𝑅[𝑥].
Theorem 9 (see [15]). If 𝛼 : 𝑅 → [0, 1] is a fuzzy ideal of 𝑅, then the fuzzy set √𝛼 is a fuzzy ideal of 𝑅. Lemma 10 (see [15]). If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then √𝛼 ∩ 𝛽 = √𝛼 ∩ √𝛽. Lemma 11. If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then √𝛼 ∪ 𝛽 = √𝛼 ∪ √𝛽. Proof. If 𝑎 is an element of 𝑅, then √𝛼 ∪ 𝛽 (𝑎) = ⋁ {(𝛼 ∪ 𝛽) (𝑎𝑛 ) | 𝑛 > 0} = ⋁ {max {𝛼 (𝑎𝑛 ) , 𝛽 (𝑎𝑛 )} | 𝑛 > 0} = max {⋁ {𝛼 (𝑎𝑛 ) | 𝑛 > 0} , ⋁ {𝛽 (𝑎𝑛 ) | 𝑛 > 0}}
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= max {√𝛼 (𝑎) , √𝛽 (𝑎)} = (√𝛼 ∪ √𝛽) (𝑎) . This proves that √𝛼 ∪ 𝛽 = √𝛼 ∪ √𝛽.
The fuzzy ideal 𝛼𝑥 discussed in Theorem 5 is called the fuzzy polynomial ideal [13] of 𝑅[𝑥] induced by a fuzzy ideal 𝛼. Theorem 6 (see [13]). Let 𝛼 : 𝑅 → [0, 1] be a fuzzy ideal of a ring 𝑅. Then 𝛼 is a fuzzy prime ideal of 𝑅 if and only if 𝛼𝑥 is a fuzzy prime ideal of 𝑅[𝑥]. Notation 1. Let 𝛼 : 𝑅 → [0, 1] be a fuzzy subset of a set 𝑅. We denote a level set 𝛼∗ by 𝛼∗ fl {𝑎 ∈ 𝑅 | 𝛼(𝑎) = 𝛼(0)}, and we know that 𝛼(0) ≧ 𝛼(𝑥) for all 𝑥 ∈ 𝑅. The set of all polynomials 𝑖 𝑓(𝑥) = ∑𝑚 𝑖=0 𝑎𝑖 𝑥 ∈ 𝑅[𝑥] whose 𝛼’s values 𝛼(𝑎𝑖 ) are equal to 𝛼(0) for all 𝑖 = 0, 1, . . . , 𝑛, is denoted by 𝛼∗ [𝑥]. Theorem 7 (see [13]). If 𝛼 and 𝛽 are fuzzy ideals of a ring 𝑅, then (i) (𝛼 ∩ 𝛽)𝑥 = 𝛼𝑥 ∩ 𝛽𝑥 , (ii) 𝛼𝑥 ∪ 𝛽𝑥 ⊆ (𝛼 ∪ 𝛽)𝑥 . Let 𝑓 : 𝑅 → 𝑅 be a homomorphism of rings. A map 𝑓𝑥 : 𝑅[𝑥] → 𝑅 [𝑥] defined by 𝑓𝑥 (𝑎0 + 𝑎1 𝑥 + ⋅ ⋅ ⋅ + 𝑎𝑛 𝑥𝑛 ) fl 𝑓(𝑎0 ) + 𝑓(𝑎1 )𝑥 + ⋅ ⋅ ⋅ + 𝑓(𝑎𝑛 )𝑥𝑛 is obviously a ring homomorphism, and we call it an induced homomorphism [13] by 𝑓. Theorem 8 (see [13]). Let 𝑓 : 𝑅 → 𝑅 be an epimorphism of rings and let 𝑓𝑥 be an induced homomorphism of 𝑓. If 𝛼 is an 𝑓-invariant fuzzy ideal of 𝑅, then (𝑓(𝛼))𝑥 = 𝑓𝑥 (𝛼𝑥 ).
3. Fuzzy Polynomial Ideals In this section, we study some relations between the radical of the fuzzy polynomial induced by a fuzzy ideal and the radical of a fuzzy ideal of a ring. A fuzzy ideal 𝛼 : 𝑅 → [0, 1] of a ring 𝑅 is called a fuzzy prime ideal [14] of 𝑅 if 𝛼∗ is a prime ideal of 𝑅. A fuzzy set √𝛼 : 𝑅 → [0, 1], defined as √𝛼(𝑎) := ⋁{𝛼(𝑎𝑛 ) | 𝑛 > 0}, is called a fuzzy nil radical [15] of 𝛼.
Since 𝛼𝑥 is a fuzzy ideal of a polynomial ring 𝑅[𝑥] by Theorem 5, the fuzzy set √𝛼𝑥 : 𝑅[𝑥] → [0, 1] is the fuzzy nil radical of 𝛼𝑥 . The following theorem gives that the two fuzzy nil radicals have the same value. Theorem 12. If 𝛼 : 𝑅 → [0, 1] is a fuzzy ideal of 𝑅, then (√𝛼𝑥 )𝑥 = (√𝛼)𝑥 .
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Proof. Let 𝑓(𝑥) fl ∑𝑚 𝑖=0 ∈ 𝑅[𝑥] be any element of 𝑅[𝑥]. Then, by Theorem 5, we have 𝛼𝑥 (𝑎𝑗𝑛 ) = 𝛼𝑥 (𝑎𝑗𝑛 +0𝑥+0𝑥2 +⋅ ⋅ ⋅+0𝑥𝑚 ) = min{𝛼(𝑎𝑗𝑛 ), 𝛼(0), . . . , 𝛼(0)} = 𝛼(𝑎𝑗𝑛 ). Since √𝛼𝑥 is a fuzzy ideal of 𝑅[𝑥], we obtain 𝑚
(√𝛼𝑥 )𝑥 (𝑓 (𝑥)) = min {√𝛼𝑥 (𝑎𝑖 )} 𝑖=0 𝑚
= min {⋁ {𝛼𝑥 (𝑎𝑖 𝑛 ) | 𝑛 > 0}} 𝑖=0 𝑚
(6) 𝑛
= min {⋁ {𝛼 (𝑎𝑖 ) | 𝑛 > 0}} 𝑖=0 𝑚
= min {√𝛼 (𝑎𝑖 )} = (√𝛼)𝑥 (𝑓 (𝑥)) . 𝑖=0
This proves that (√𝛼)𝑥 = (√𝛼𝑥 )𝑥 . Theorem 13. If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then (√𝛼 ∩ 𝛽) = (√𝛼)𝑥 ∩ (√𝛽) . 𝑥
𝑥
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Proof. If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then 𝛼𝑥 and 𝛽𝑥 are fuzzy ideals of 𝑅[𝑥] by Theorem 5. It follows from Theorems 12 and 7(i) and Lemma 10 that
Proof. Given a polynomial 𝑔(𝑥) = 𝑏0 +𝑏1 𝑥+⋅ ⋅ ⋅+𝑏𝑚 𝑥𝑚 ∈ 𝑅[𝑥], we have (√𝑓−1 (𝛽)) (𝑔 (𝑥)) = min {√𝑓−1 𝛽 (𝑏0 ) , 𝑥
(√𝛼 ∩ 𝛽) = (√(𝛼 ∩ 𝛽)𝑥 ) 𝑥
= (√𝛼𝑥 ∩ 𝛽𝑥 )
[Theorem 12]
𝑥
√𝑓−1 𝛽 (𝑏1 ) , . . . , √𝑓−1 𝛽 (𝑏𝑚 )} = min {⋁ {𝑓−1 𝛽 (𝑏0 𝑛 ) | 𝑛 > 0} ,
[Theorem 7 (i)]
𝑥
= (√𝛼𝑥 ∩ √𝛽𝑥 )
[Lemma 10]
𝑥
= (√𝛼𝑥 )𝑥 ∩ (√𝛽𝑥 )
(8)
(12)
= min {⋁ {𝛽 (𝑓 (𝑏0 ) ) | 𝑛 > 0} , 𝑛
⋁ {𝛽 (𝑓 (𝑏1 ) ) | 𝑛 > 0} , . . . ,
[Theorem 12]
𝑥
⋁ {𝑓−1 𝛽 (𝑏𝑚 𝑛 ) | 𝑛 > 0}} 𝑛
[Theorem 7 (i)]
𝑥
= (√𝛼)𝑥 ∩ (√𝛽)
⋁ {𝑓−1 𝛽 (𝑏1 𝑛 ) | 𝑛 > 0} , . . . ,
𝑛
⋁ {𝛽 (𝑓 (𝑏𝑚 ) ) | 𝑛 > 0}} = min {√𝛽 (𝑓 (𝑏0 )) ,
proving the theorem.
√𝛽 (𝑓 (𝑏1 )) , . . . , √𝛽 (𝑓 (𝑏𝑚 ))} = (√𝛽)
Theorem 14. If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then
𝑥
⋅ (𝑓𝑥 (𝑔 (𝑥))) = 𝑓𝑥 −1 [(√𝛽) ] (𝑔 (𝑥)) . (√𝛼)𝑥 ∪ (√𝛽) ⊆ (√𝛼 ∪ 𝛽) . 𝑥
𝑥
𝑥
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Proof. Since 𝛼𝑥 and 𝛽𝑥 are fuzzy ideals of 𝑅[𝑥] by Theorem 5, we obtain (√𝛼)𝑥 ∪ (√𝛽) = (√𝛼𝑥 )𝑥 ∪ (√𝛽𝑥 ) 𝑥
𝑥
[Theorem 7 (ii)] = (√𝛼𝑥 ∪ 𝛽𝑥 )
⊆ (√(𝛼 ∪ 𝛽)𝑥 ) = (√𝛼 ∪ 𝛽)
𝑥
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[Lemma 11]
𝑥
𝑥
Proposition 16 (see [15]). Let 𝑓 : 𝑅 → 𝑅 be a ring epimorphism from 𝑅 onto 𝑅 , and let 𝛼 : 𝑅 → [0, 1] be a fuzzy ideal of 𝑅. If 𝛼 is constant on Ker 𝑓, then 𝑓(√𝛼) = √𝑓(𝛼). Theorem 17. Let 𝑓 : 𝑅 → 𝑅 be a homomorphism of rings and let 𝑓𝑥 be the induced homomorphism of 𝑓. If a fuzzy ideal 𝛼 of 𝑅 is constant on Ker 𝑓, then the fuzzy polynomial ideal 𝛼𝑥 is constant on Ker 𝑓𝑥 .
𝑥
[Theorem 12] ⊆ (√𝛼𝑥 ∪ √𝛽𝑥 )
This proves that 𝑓𝑥 −1 [(√𝛽)𝑥 ] = (√𝑓−1 (𝛽))𝑥 .
Proof. Let 𝛼(𝑎) = 𝑘0 for all 𝑎 ∈ Ker 𝑓 and let 𝑔(𝑥) = 𝑏0 + 𝑏1 𝑥 + ⋅ ⋅ ⋅ + 𝑏𝑚 𝑥𝑚 be any element of Ker 𝑓𝑥 . Then 0 = 𝑓𝑥 (𝑔(𝑥)) = 𝑓(𝑏0 ) + 𝑓(𝑏1 )𝑥 + ⋅ ⋅ ⋅ + 𝑓(𝑏𝑚 )𝑥𝑚 . It follows that 𝑓(𝑏𝑖 ) = 0 for all 𝑖 = 0, 1, . . . , 𝑚. Hence 𝑏𝑖 ∈ Ker 𝑓 for all 𝑖 = 0, 1, . . . , 𝑚; that is, 𝛼(𝑏𝑖 ) = 𝑘0 for all 𝑖 = 0, 1, . . . , 𝑚. This shows that 𝛼𝑥 (𝑔(𝑥)) = min{𝛼(𝑏0 ), 𝛼(𝑏1 ), . . . , 𝛼(𝑏𝑚 )} = 𝑘0 , proving the theorem. Corollary 18. Let 𝑓 : 𝑅 → 𝑅 be an epimorphism of rings and let 𝑓𝑥 be the induced homomorphism of 𝑓. If an 𝑓-invariant fuzzy ideal 𝛼 of 𝑅 is constant on Ker 𝑓, then
[Theorem 7 (ii)]
[Theorem 12] .
𝑓𝑥 (√𝛼𝑥 ) = √(𝑓 (𝛼))𝑥 .
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Proof. It follows from Proposition 16 and Theorem 8 that 𝑓𝑥 (√𝛼𝑥 ) = √𝑓𝑥 (𝛼𝑥 ) = √(𝑓(𝛼))𝑥 .
This proves that (√𝛼)𝑥 ∪ (√𝛽)𝑥 ⊆ (√𝛼 ∪ 𝛽)𝑥 . Theorem 15. Let 𝛽 be a fuzzy ideal of 𝑅 and let 𝑓 : 𝑅 → 𝑅 be a homomorphism of rings. If 𝑓𝑥 is the induced homomorphism of 𝑓, that is, 𝑓𝑥 (∑𝑛𝑖=0 𝑎𝑖 𝑥𝑖 ) = ∑𝑛𝑖=0 𝑓(𝑎𝑖 )𝑥𝑖 , then
Theorem 19. Let 𝛼 be a fuzzy ideal of 𝑅 and let 𝛼𝑥 be its fuzzy polynomial ideal of 𝑅[𝑥]. If 𝛽 is a fuzzy prime ideal of 𝑅[𝑥] such that 𝛼𝑥 ⫅ 𝛽, then there exists a fuzzy prime ideal 𝛼0 of 𝑅 such that (𝛼0 )∗ = 𝛽∗ ∩ 𝑅 and 𝛼 ⫅ 𝛼0 .
𝑓𝑥 −1 [(√𝛽) ] = (√𝑓−1 (𝛽)) .
Proof. Since 𝛽 is a fuzzy prime ideal of 𝑅[𝑥], 𝛽∗ is a prime ideal of 𝑅[𝑥]. If we define 𝛾 := 𝛽∗ ∩ 𝑅, then it is easy to show
𝑥
𝑥
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that 𝛾 is a prime ideal of 𝑅. Define a fuzzy subset 𝛼0 : 𝑅 → [0, 1] by {𝛽 (0) 𝛼0 (𝑎) = { 𝛽 { (𝑎)
if 𝑎 ∈ 𝛾, if 𝑎 ∉ 𝛾.
Corollary 23. Let 𝛼 be a fuzzy ideal of 𝑅 and let 𝛼𝑥 be its fuzzy polynomial ideal of 𝑅[𝑥]. If the map 𝜙 defined in Theorem 22 is an onto map, then (𝑟 (𝛼))𝑥 = 𝑟 (𝛼𝑥 ) .
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Then, by routine calculations, we show that 𝛼0 is a fuzzy ideal of 𝑅 satisfying (𝛼0 )∗ = 𝛾. We claim that 𝛼 ⫅ 𝛼0 . Given 𝑎 ∈ 𝑅, if 𝑎 ∈ 𝛾, then 𝛼(𝑎) = 𝛼𝑥 (𝑎) ≤ 𝛽(𝑎) ≤ 𝛽(0) = 𝛼0 (𝑎). If 𝑎 ∉ 𝛾, then 𝛼(𝑎) = 𝛼𝑥 (𝑎) ≤ 𝛽(𝑎) = 𝛼0 (𝑎). Since 𝛾 is a prime ideal of 𝑅, (𝛼0 )∗ is a prime ideal of 𝑅. This shows that 𝛼0 is a fuzzy prime ideal of 𝑅, proving the theorem.
Proof. If 𝛽 is any element of FPI(𝛼𝑥 ), then there exists 𝛾 ∈ FPI(𝛼) such that 𝛾𝑥 = 𝜙(𝛾) = 𝛽 with 𝛼 ⊆ 𝛾. Thus (𝑟(𝛼))𝑥 = 𝑟(𝛼𝑥 ). This shows that the reverse inclusion in Theorem 21 holds. Example 24. Let Z be set of all integers. Let {1 𝛼 (𝑥) fl { 0 {
Definition 20 (see [15]). Let 𝛼 : 𝑅 → [0, 1] be a fuzzy ideal of 𝑅. The fuzzy ideal 𝑟(𝛼) defined by 𝑟 (𝛼) fl ⋂ {𝛽 | 𝛼 ⫅ 𝛽, 𝛽 : a fuzzy prime ideal of 𝑅}
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is called the prime fuzzy radical of 𝛼. Theorem 21. Let 𝛼 be a fuzzy ideal of 𝑅 and let 𝛼𝑥 be its fuzzy polynomial ideal of 𝑅[𝑥]. Then 𝑟 (𝛼𝑥 ) ⊆ (𝑟 (𝛼))𝑥 .
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Proof. By Theorem 6, 𝛽𝑖 is a fuzzy prime ideal of 𝑅 with 𝛼 ⊆ 𝛽𝑖 if and only if (𝛽𝑖 )𝑥 is a fuzzy prime ideal of 𝑅[𝑥] with 𝛼𝑥 ⊆ (𝛽𝑖 )𝑥 . It follows from Theorem 7(i) that
(20)
if 𝑥 ∈ 2Z, if 𝑥 ∉ 2Z.
(21)
Then 𝛼 is a fuzzy prime ideal of Z, since 𝛼∗ = 2Z is a prime ideal of Z, and its induced polynomial ideal 𝛼𝑥 is {1 if 𝑓 (𝑥) ∈ 2Z [𝑥] , 𝛼𝑥 (𝑓 (𝑥)) fl { 0 if 𝑓 (𝑥) ∉ 2Z [𝑥] . {
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By Theorem 6, the fuzzy polynomial ideal 𝛼𝑥 induced by 𝛼 is a fuzzy prime ideal of Z[𝑥]. Hence (𝑟(𝛼))𝑥 = 𝛼𝑥 = 𝑟(𝛼𝑥 ).
Competing Interests The authors declare that they have no competing interests.
(𝑟 (𝛼))𝑥 = (⋂ {𝛽𝑖 | 𝛼 ⊆ 𝛽𝑖 , 𝛽𝑖 is a fuzzy prime ideal of 𝑅})𝑥
Acknowledgments
= (⋂ {(𝛽𝑖 )𝑥 | 𝛼
This research was supported by Hallym University Research Fund, 2015 (HRF-201503-016).
⊆ 𝛽𝑖 , 𝛽𝑖 is a fuzzy prime ideal of 𝑅}) = ⋂ {(𝛽𝑖 )𝑥 | (17)
References
𝛼𝑥 ⊆ (𝛽𝑖 )𝑥 , (𝛽𝑖 )𝑥 is a fuzzy prime ideal of 𝑅 [𝑥]} ⊇ ⋂ {𝛾𝑖 | 𝛼𝑥 ⊆ 𝛾𝑖 , 𝛾𝑖 is a fuzzy prime ideal of 𝑅 [𝑥]} = 𝑟 (𝛼𝑥 ) ,
proving the theorem. Notation 2. Let 𝛼 be a fuzzy ideal of 𝑅 and let 𝛼𝑥 be its fuzzy polynomial ideal of 𝑅[𝑥]. We denote FPI(𝛼) by FPI (𝛼) fl {𝛽 | 𝛼 ⊆ 𝛽, 𝛽 is a fuzzy prime ideal of 𝑅} (18) and FPI(𝛼𝑥 ) by FPI (𝛼𝑥 ) fl {𝛾 | 𝛼𝑥 ⊆ 𝛾, 𝛾 is a fuzzy prime ideal of 𝑅 [𝑥]} .
(19)
Theorem 22. Let 𝛼 be a fuzzy ideal of 𝑅 and let 𝛼𝑥 be its fuzzy polynomial ideal of 𝑅[𝑥]. Then a map 𝜙 : FPI(𝛼) → FPI(𝛼𝑥 ) defined by 𝜙(𝛽) fl 𝛽𝑥 is one-one. Proof. If 𝛽, 𝛾 ∈ FPI(𝛼) such that 𝜙(𝛽) = 𝜙(𝛾), then 𝛽𝑥 = 𝛾𝑥 . It follows that 𝛽𝑥 (𝑎) = 𝛾𝑥 (𝑎) for all 𝑎 ∈ 𝑅, and hence 𝛽(𝑎) = 𝛾(𝑎) for all 𝑎 ∈ 𝑅, proving that 𝛽 = 𝛾. Hence 𝜙 is one-one.
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