research article radical structures of fuzzy polynomial ... · of a fuzzy ideal of a ring. a fuzzy...

Research ArticleRadical Structures of Fuzzy Polynomial Ideals in a Ring

Hee Sik Kim,1 Chang Bum Kim,2 and Keum Sook So3

1Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea2Department of Mathematics, Kookmin University, Seoul 136-702, Republic of Korea3Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea

Correspondence should be addressed to Keum Sook So; [email protected]

Received 13 November 2015; Accepted 28 February 2016

Academic Editor: Cengiz Çinar

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 afuzzy ideal 𝛽 of 𝑅 and a homomorphism 𝑓 : 𝑅 → 𝑅, we show that if 𝑓

𝑥is the induced homomorphism of 𝑓, that is, 𝑓

𝑥(∑𝑛

𝑖=0𝑎𝑖𝑥𝑖) =

∑𝑛

𝑖=0𝑓(𝑎𝑖)𝑥𝑖, then 𝑓

𝑥

−1[(√𝛽)𝑥] = (√𝑓−1(𝛽))

𝑥.

1. Introduction

Zadeh [1] introduced the notion of a fuzzy subset 𝐴 of a set𝑋 as a function from 𝑋 into [0, 1]. Rosenfeld [2] appliedthis 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 examinedthe fuzzy prime ideals of a ring. The concept of fuzzy idealswas applied to several algebras: 𝐵𝑁-algebras [5], 𝐵𝐿-algebras[6], semirings [7], and semigroups [8]. Ersoy et al. [9] appliedthe concept of intuitionistic fuzzy soft sets to rings, and Shahet al. [10] discussed intuitionistic fuzzy normal subrings overa nonassociative ring. Prajapati [11] investigated residual ofideals of an 𝐿-ring. Dheena and Mohanraj [12] obtained acondition for a fuzzy small right ideal to be fuzzy small primeright ideal.

The present authors [13] introduced the notion of a fuzzypolynomial ideal 𝛼

𝑥of a polynomial ring 𝑅[𝑥] induced by

a fuzzy ideal 𝛼 of a ring 𝑅 and obtained an isomorphismtheorem of a ring of fuzzy cosets of 𝛼

𝑥. It was shown that a

fuzzy ideal 𝛼 of a ring is fuzzy prime if and only if 𝛼𝑥is a

fuzzy prime ideal of 𝑅[𝑥]. Moreover, we showed that if 𝛼𝑥is a

fuzzy maximal ideal of 𝑅[𝑥], then 𝛼 is a fuzzy maximal idealof 𝑅.

In this paper we investigate the radical structure of a fuzzypolynomial ideal induced by a fuzzy ideal of a ring and studytheir properties.

2. Preliminaries

In this section, we review some definitions which will beused in the later section.Throughout this paper unless statedotherwise all rings are commutative rings with identity.

Definition 1 (see [3]). A fuzzy ideal of a ring 𝑅 is a function𝛼 : 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 ofrings and let 𝛽 be a fuzzy subset of 𝑆. We define a fuzzy subset𝑓−1𝛽 of 𝑅 by 𝑓−1𝛽(𝑥) fl 𝛽(𝑓(𝑥)) for all 𝑥 ∈ 𝑅.

Definition 3 (see [2]). Let 𝑓 : 𝑅 → 𝑆 be a homomorphism ofrings and let 𝛼 be a fuzzy subset of𝑅. We define a fuzzy subset𝑓(𝛼) of 𝑆 by

𝑓 (𝛼) (𝑦)

fl{{{

sup {𝛼 (𝑡) | 𝑡 ∈ 𝑅, 𝑓 (𝑡) = 𝑦} if 𝑓−1 (𝑦) ̸= 0,

0 if 𝑓−1 (𝑦) = 0.

(1)

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016, Article ID 7821678, 5 pageshttp://dx.doi.org/10.1155/2016/7821678

2 Discrete Dynamics in Nature and Society

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-invariantif 𝑓(𝑥) = 𝑓(𝑦) implies 𝛼(𝑥) = 𝛼(𝑦), where 𝑥, 𝑦 ∈ 𝑅.

Zadeh [1] defined the following notions.The union of twofuzzy subsets 𝛼 and 𝛽 of a set 𝑆, denoted by 𝛼 ∪ 𝛽, is a fuzzysubset of 𝑆 defined by

(𝛼 ∪ 𝛽) (𝑥) fl max {𝛼 (𝑥) , 𝛽 (𝑥)} (2)

for all 𝑥 ∈ 𝑆.The intersection of 𝛼 and𝛽, symbolized by 𝛼∩𝛽, is a fuzzy

subset of 𝑆, defined by

(𝛼 ∩ 𝛽) (𝑥) fl min {𝛼 (𝑥) , 𝛽 (𝑥)} (3)

for all 𝑥 ∈ 𝑆.

Theorem 5 (see [13]). Let 𝛼 : 𝑅 → [0, 1] be a fuzzy ideal ofa ring 𝑅 and let 𝑓(𝑥) = 𝑎

0+ 𝑎1𝑥 + 𝑎2𝑥2 + ⋅ ⋅ ⋅ + 𝑎

𝑛𝑥𝑛 be a

polynomial in 𝑅[𝑥]. Define a fuzzy set 𝛼𝑥: 𝑅[𝑥] → [0, 1] by

𝛼𝑥(𝑓(𝑥)) fl min

𝑖{𝛼(𝑎𝑖) | 𝑎𝑖’s are coefficients of 𝑓(𝑥)}. Then

𝛼𝑥is a fuzzy ideal of 𝑅[𝑥].

The fuzzy ideal 𝛼𝑥discussed 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 aring 𝑅. Then 𝛼 is a fuzzy prime ideal of 𝑅 if and only if 𝛼

𝑥is a

fuzzy prime ideal of 𝑅[𝑥].

Notation 1. Let 𝛼 : 𝑅 → [0, 1] be a fuzzy subset of a set 𝑅. Wedenote a level set 𝛼

∗by 𝛼∗fl {𝑎 ∈ 𝑅 | 𝛼(𝑎) = 𝛼(0)}, and we

know that𝛼(0) ≧ 𝛼(𝑥) for all 𝑥 ∈ 𝑅.The set of all polynomials𝑓(𝑥) = ∑

𝑚

𝑖=0𝑎𝑖𝑥𝑖 ∈ 𝑅[𝑥] whose 𝛼’s values 𝛼(𝑎

𝑖) 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+ 𝑎1𝑥 + ⋅ ⋅ ⋅ + 𝑎

𝑛𝑥𝑛) fl 𝑓(𝑎

0) +

𝑓(𝑎1)𝑥 + ⋅ ⋅ ⋅ + 𝑓(𝑎

𝑛)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 ofrings and let 𝑓

𝑥be 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 ofthe fuzzy polynomial induced by a fuzzy ideal and the radicalof a fuzzy ideal of a ring.

A fuzzy ideal 𝛼 : 𝑅 → [0, 1] of a ring 𝑅 is called a fuzzyprime 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}, iscalled a fuzzy nil radical [15] of 𝛼.

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}}

= max {√𝛼 (𝑎) , √𝛽 (𝑎)} = (√𝛼 ∪ √𝛽) (𝑎) .

(4)

This proves that√𝛼 ∪ 𝛽 = √𝛼 ∪ √𝛽.

Since 𝛼𝑥is a fuzzy ideal of a polynomial ring 𝑅[𝑥] by

Theorem 5, the fuzzy set √𝛼𝑥 : 𝑅[𝑥] → [0, 1] is the fuzzy nilradical of 𝛼

𝑥. 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 . (5)

Proof. Let 𝑓(𝑥) fl ∑𝑚𝑖=0

∈ 𝑅[𝑥] be any element of 𝑅[𝑥]. Then,byTheorem 5, we have𝛼

𝑥(𝑎𝑛𝑗) = 𝛼𝑥(𝑎𝑛𝑗+0𝑥+0𝑥2+⋅ ⋅ ⋅+0𝑥𝑚) =

min{𝛼(𝑎𝑛𝑗), 𝛼(0), . . . , 𝛼(0)} = 𝛼(𝑎𝑛

𝑗). Since√𝛼𝑥 is a fuzzy ideal

of 𝑅[𝑥], we obtain

(√𝛼𝑥)𝑥 (𝑓 (𝑥)) =𝑚

min𝑖=0

{√𝛼𝑥 (𝑎𝑖)}

=𝑚

min𝑖=0

{⋁{𝛼𝑥(𝑎𝑖

𝑛) | 𝑛 > 0}}

=𝑚

min𝑖=0

{⋁{𝛼 (𝑎𝑖

𝑛) | 𝑛 > 0}}

=𝑚

min𝑖=0

{√𝛼 (𝑎𝑖)} = (√𝛼)

𝑥(𝑓 (𝑥)) .

(6)

This proves that (√𝛼)𝑥= (√𝛼𝑥)𝑥.

Theorem 13. If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then

(√𝛼 ∩ 𝛽)𝑥

= (√𝛼)𝑥∩ (√𝛽)

𝑥

. (7)

Discrete Dynamics in Nature and Society 3

Proof. If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then 𝛼𝑥and 𝛽

𝑥are

fuzzy ideals of 𝑅[𝑥] by Theorem 5. It follows fromTheorems12 and 7(i) and Lemma 10 that

(√𝛼 ∩ 𝛽)𝑥

= (√(𝛼 ∩ 𝛽)𝑥)𝑥

[Theorem 12]

= (√𝛼𝑥∩ 𝛽𝑥)𝑥

[Theorem 7 (i)]

= (√𝛼𝑥 ∩ √𝛽𝑥)𝑥

[Lemma 10]

= (√𝛼𝑥)𝑥 ∩ (√𝛽𝑥)𝑥

[Theorem 7 (i)]

= (√𝛼)𝑥∩ (√𝛽)

𝑥

[Theorem 12]

(8)

proving the theorem.

Theorem 14. If 𝛼 and 𝛽 are fuzzy ideals of 𝑅, then

(√𝛼)𝑥∪ (√𝛽)

𝑥

⊆ (√𝛼 ∪ 𝛽)𝑥

. (9)

Proof. Since 𝛼𝑥and 𝛽

𝑥are fuzzy ideals of 𝑅[𝑥] byTheorem 5,

we obtain

(√𝛼)𝑥∪ (√𝛽)

𝑥

= (√𝛼𝑥)𝑥 ∪ (√𝛽𝑥)𝑥

[Theorem 12]

⊆ (√𝛼𝑥 ∪ √𝛽𝑥)𝑥

[Theorem 7 (ii)]

= (√𝛼𝑥∪ 𝛽𝑥)𝑥

[Lemma 11]

⊆ (√(𝛼 ∪ 𝛽)𝑥)𝑥

[Theorem 7 (ii)]

= (√𝛼 ∪ 𝛽)𝑥

[Theorem 12] .

(10)

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 bea homomorphism of rings. If 𝑓

𝑥is the induced homomorphism

of 𝑓, that is, 𝑓𝑥(∑𝑛

𝑖=0𝑎𝑖𝑥𝑖) = ∑

𝑛

𝑖=0𝑓(𝑎𝑖)𝑥𝑖, then

𝑓𝑥

−1 [(√𝛽)𝑥

] = (√𝑓−1 (𝛽))𝑥

. (11)

Proof. Given a polynomial𝑔(𝑥) = 𝑏0+𝑏1𝑥+⋅ ⋅ ⋅+𝑏

𝑚𝑥𝑚 ∈ 𝑅[𝑥],

we have

(√𝑓−1 (𝛽))𝑥

(𝑔 (𝑥)) = min {√𝑓−1𝛽 (𝑏0) ,

√𝑓−1𝛽 (𝑏1) , . . . , √𝑓−1𝛽 (𝑏

𝑚)}

= min {⋁{𝑓−1𝛽 (𝑏0

𝑛) | 𝑛 > 0} ,

⋁{𝑓−1𝛽 (𝑏1

𝑛) | 𝑛 > 0} , . . . ,

⋁ {𝑓−1𝛽 (𝑏𝑚

𝑛) | 𝑛 > 0}}

= min {⋁{𝛽 (𝑓 (𝑏0)𝑛

) | 𝑛 > 0} ,

⋁{𝛽 (𝑓 (𝑏1)𝑛

) | 𝑛 > 0} , . . . ,

⋁ {𝛽 (𝑓 (𝑏𝑚)𝑛

) | 𝑛 > 0}} = min {√𝛽 (𝑓 (𝑏0)) ,

√𝛽 (𝑓 (𝑏1)) , . . . , √𝛽 (𝑓 (𝑏

𝑚))} = (√𝛽)

𝑥

⋅ (𝑓𝑥(𝑔 (𝑥))) = 𝑓

𝑥

−1 [(√𝛽)𝑥

] (𝑔 (𝑥)) .

(12)

This proves that 𝑓𝑥

−1[(√𝛽)𝑥] = (√𝑓−1(𝛽))

𝑥.

Proposition 16 (see [15]). Let 𝑓 : 𝑅 → 𝑅 be a ringepimorphism from 𝑅 onto 𝑅, and let 𝛼 : 𝑅 → [0, 1] be a fuzzyideal of 𝑅. If 𝛼 is constant on Ker𝑓, then 𝑓(√𝛼) = √𝑓(𝛼).

Theorem 17. Let𝑓 : 𝑅 → 𝑅 be a homomorphism of rings andlet 𝑓𝑥be the induced homomorphism of 𝑓. If a fuzzy ideal 𝛼 of

𝑅 is constant on Ker𝑓, then the fuzzy polynomial ideal 𝛼𝑥is

constant on Ker𝑓𝑥.

Proof. Let 𝛼(𝑎) = 𝑘0for all 𝑎 ∈ Ker𝑓 and let 𝑔(𝑥) = 𝑏

0+𝑏1𝑥+

⋅ ⋅ ⋅ + 𝑏𝑚𝑥𝑚 be any element of Ker𝑓

𝑥. Then 0 = 𝑓

𝑥(𝑔(𝑥)) =

𝑓(𝑏0) +𝑓(𝑏

1)𝑥+ ⋅ ⋅ ⋅ +𝑓(𝑏

𝑚)𝑥𝑚. It follows that 𝑓(𝑏

𝑖) = 0 for all

𝑖 = 0, 1, . . . , 𝑚. Hence 𝑏𝑖∈ Ker𝑓 for all 𝑖 = 0, 1, . . . , 𝑚; that is,

𝛼(𝑏𝑖) = 𝑘0for all 𝑖 = 0, 1, . . . , 𝑚. This shows that 𝛼

𝑥(𝑔(𝑥)) =

min{𝛼(𝑏0), 𝛼(𝑏1), . . . , 𝛼(𝑏

𝑚)} = 𝑘

0, proving the theorem.

Corollary 18. Let 𝑓 : 𝑅 → 𝑅 be an epimorphism of rings andlet 𝑓𝑥be the induced homomorphism of 𝑓. If an 𝑓-invariant

fuzzy ideal 𝛼 of 𝑅 is constant on Ker𝑓, then

𝑓𝑥(√𝛼𝑥) = √(𝑓 (𝛼))𝑥. (13)

Proof. It follows from Proposition 16 and Theorem 8 that𝑓𝑥(√𝛼𝑥) = √𝑓𝑥(𝛼𝑥) = √(𝑓(𝛼))𝑥.

Theorem 19. Let 𝛼 be a fuzzy ideal of 𝑅 and let 𝛼𝑥be its fuzzy

polynomial ideal of 𝑅[𝑥]. If 𝛽 is a fuzzy prime ideal of 𝑅[𝑥]such that 𝛼

𝑥⫅ 𝛽, then there exists a fuzzy prime ideal 𝛼

0of 𝑅

such that (𝛼0)∗= 𝛽∗∩ 𝑅 and 𝛼 ⫅ 𝛼

0.

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

4 Discrete Dynamics in Nature and Society

that 𝛾 is a prime ideal of 𝑅. Define a fuzzy subset 𝛼0: 𝑅 →

[0, 1] by

𝛼0(𝑎) =

{{{

𝛽 (0) if 𝑎 ∈ 𝛾,

𝛽 (𝑎) if 𝑎 ∉ 𝛾.(14)

Then, by routine calculations, we show that 𝛼0is a fuzzy ideal

of 𝑅 satisfying (𝛼0)∗= 𝛾. We claim that 𝛼 ⫅ 𝛼

0. Given 𝑎 ∈ 𝑅,

if 𝑎 ∈ 𝛾, then 𝛼(𝑎) = 𝛼𝑥(𝑎) ≤ 𝛽(𝑎) ≤ 𝛽(0) = 𝛼

0(𝑎). If 𝑎 ∉ 𝛾,

then 𝛼(𝑎) = 𝛼𝑥(𝑎) ≤ 𝛽(𝑎) = 𝛼

0(𝑎). Since 𝛾 is a prime ideal

of 𝑅, (𝛼0)∗is a prime ideal of 𝑅. This shows that 𝛼

0is a fuzzy

prime ideal of 𝑅, proving the theorem.

Definition 20 (see [15]). Let 𝛼 : 𝑅 → [0, 1] be a fuzzy ideal of𝑅. The fuzzy ideal 𝑟(𝛼) defined by

𝑟 (𝛼) fl ⋂{𝛽 | 𝛼 ⫅ 𝛽, 𝛽 : a fuzzy prime ideal of 𝑅} (15)

is called the prime fuzzy radical of 𝛼.


polynomial ideal of 𝑅[𝑥]. Then

𝑟 (𝛼𝑥) ⊆ (𝑟 (𝛼))

𝑥. (16)

Proof. ByTheorem 6,𝛽𝑖is a fuzzy prime ideal of𝑅with𝛼 ⊆ 𝛽

𝑖

if and only if (𝛽𝑖)𝑥is a fuzzy prime ideal of 𝑅[𝑥] with 𝛼

𝑥⊆

(𝛽𝑖)𝑥. It follows fromTheorem 7(i) that

(𝑟 (𝛼))𝑥= (⋂{𝛽

𝑖| 𝛼

⊆ 𝛽𝑖, 𝛽𝑖is a fuzzy prime ideal of 𝑅})

𝑥

= (⋂{(𝛽𝑖)𝑥| 𝛼

⊆ 𝛽𝑖, 𝛽𝑖is a fuzzy prime ideal of 𝑅}) = ⋂{(𝛽

𝑖)𝑥|

𝛼𝑥⊆ (𝛽𝑖)𝑥, (𝛽𝑖)𝑥is a fuzzy prime ideal of 𝑅 [𝑥]}

⊇ ⋂{𝛾𝑖| 𝛼𝑥⊆ 𝛾𝑖,

𝛾𝑖is a fuzzy prime ideal of 𝑅 [𝑥]} = 𝑟 (𝛼

𝑥) ,

(17)

proving the theorem.

Notation 2. Let 𝛼 be a fuzzy ideal of 𝑅 and let 𝛼𝑥be 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)


polynomial ideal of 𝑅[𝑥]. Then a map 𝜙 : FPI(𝛼) → FPI(𝛼𝑥)

defined by 𝜙(𝛽) fl 𝛽𝑥is one-one.

Proof. If 𝛽, 𝛾 ∈ FPI(𝛼) such that 𝜙(𝛽) = 𝜙(𝛾), then 𝛽𝑥= 𝛾𝑥. It

follows that𝛽𝑥(𝑎) = 𝛾

𝑥(𝑎) for all 𝑎 ∈ 𝑅, and hence𝛽(𝑎) = 𝛾(𝑎)

for all 𝑎 ∈ 𝑅, proving that 𝛽 = 𝛾. Hence 𝜙 is one-one.

Corollary 23. Let 𝛼 be a fuzzy ideal of𝑅 and let 𝛼𝑥be its fuzzy

polynomial ideal of 𝑅[𝑥]. If the map 𝜙 defined in Theorem 22is an onto map, then

(𝑟 (𝛼))𝑥= 𝑟 (𝛼

𝑥) . (20)

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

𝛼 (𝑥) fl{{{

1 if 𝑥 ∈ 2Z,

0 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 𝛼𝑥is

𝛼𝑥(𝑓 (𝑥)) fl

{{{

1 if 𝑓 (𝑥) ∈ 2Z [𝑥] ,

0 if 𝑓 (𝑥) ∉ 2Z [𝑥] .(22)

ByTheorem 6, the fuzzy polynomial ideal 𝛼𝑥induced 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.

Acknowledgments

This research was supported by Hallym University ResearchFund, 2015 (HRF-201503-016).

References

[1] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3,pp. 338–353, 1965.

[2] A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysisand Applications, vol. 35, pp. 512–517, 1971.

[3] W. J. Liu, “Fuzzy invariant subgroups and fuzzy ideals,” FuzzySets and Systems, vol. 8, no. 2, pp. 133–139, 1982.

[4] T. K. Mukherjee and M. K. Sen, “On fuzzy ideals of a ring I,”Fuzzy Sets and Systems, vol. 21, no. 1, pp. 99–104, 1987.

[5] G. Dymek and A. Walendziak, “(Fuzzy) ideals of BN-algebras,”The Scientific World Journal, vol. 2015, Article ID 925040, 9pages, 2015.

[6] B. L. Meng and X. L. Xin, “On fuzzy ideals of BL-algebras,”TheScientific World Journal, vol. 2014, Article ID 757382, 12 pages,2014.

[7] H. V. Kumbhojkar, “Proper fuzzification of prime ideals of ahemiring,” Advances in Fuzzy Systems, vol. 2012, Article ID801650, 8 pages, 2012.

[8] M. Khan, F. Feng, andM. N. A. Khan, “Onminimal fuzzy idealsof semigroups,” Journal of Mathematics, vol. 2013, Article ID475190, 5 pages, 2013.

[9] B. A. Ersoy, S. Onar, K. Hila, and B. Davvaz, “Some properties ofintuitionistic fuzzy soft rings,” Journal of Mathematics, vol. 2013,Article ID 650480, 8 pages, 2013.

Discrete Dynamics in Nature and Society 5

[10] T. Shah, N. Kausar, and I. Rehman, “Intuitionistic fuzzy nor-mal subring over a non-associative ring,” Analele Universitatii‘Ovidius’ Constanta, vol. 20, no. 1, pp. 369–386, 2012.

[11] A. S. Prajapati, “Residual of ideals of an L-ring,” Iranian Journalof Fuzzy Systems, vol. 4, no. 2, pp. 69–82, 2007.

[12] P. Dheena and G. Mohanraj, “Fuzzy small right ideal of rings,”Journal of Hyperstructures, vol. 2, pp. 8–17, 2013.

[13] C. B. Kim, H. S. Kim, and K. S. So, “On the fuzzy polynomialideals,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 1, pp.487–494, 2014.

[14] H. V. Kumbhojkar andM. S. Bapat, “Not-so-fuzzy fuzzy ideals,”Fuzzy Sets and Systems, vol. 37, no. 2, pp. 237–243, 1990.

[15] H. V. Kumbhojkar and M. S. Bapat, “On prime and primaryfuzzy ideals and their radicals,” Fuzzy Sets and Systems, vol. 53,no. 2, pp. 203–216, 1993.

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