Radical Structures of Fuzzy Polynomial Ideals in a Ring

Abstract

We investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study its properties. Given a fuzzy ideal of and a homomorphism , we show that if is the induced homomorphism of , that is, , then .

1. Introduction

Zadeh [1] introduced the notion of a fuzzy subset of a set as a function from into . 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: -algebras [5], -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 -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 of a polynomial ring induced by a fuzzy ideal of a ring and obtained an isomorphism theorem of a ring of fuzzy cosets of . It was shown that a fuzzy ideal of a ring is fuzzy prime if and only if is a fuzzy prime ideal of . Moreover, we showed that if is a fuzzy maximal ideal of , then is a fuzzy maximal ideal of .

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.

2. Preliminaries

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 is a function satisfying the following axioms:(1).(2).(3)for any .

Definition 2 (see [2]). Let be a homomorphism of rings and let be a fuzzy subset of . We define a fuzzy subset of by for all .

Definition 3 (see [2]). Let be a homomorphism of rings and let be a fuzzy subset of . We define a fuzzy subset of by

Definition 4 (see [2]). Let and be any sets and let be a function. A fuzzy subset of is called an -invariant if implies , where .

Zadeh [1] defined the following notions. The union of two fuzzy subsets and of a set , denoted by , is a fuzzy subset of defined byfor all .

The intersection of and , symbolized by , is a fuzzy subset of , defined byfor all .

Theorem 5 (see [13]). Let be a fuzzy ideal of a ring and let be a polynomial in . Define a fuzzy set by . Then is a fuzzy ideal of .

The fuzzy ideal discussed in Theorem 5 is called the fuzzy polynomial ideal [13] of induced by a fuzzy ideal .

Theorem 6 (see [13]). Let be a fuzzy ideal of a ring . Then is a fuzzy prime ideal of if and only if is a fuzzy prime ideal of .

Notation 1. Let be a fuzzy subset of a set . We denote a level set by , and we know that for all . The set of all polynomials whose ’s values are equal to for all , is denoted by .

Theorem 7 (see [13]). If and are fuzzy ideals of a ring , then(i),(ii)

Let be a homomorphism of rings. A map defined by is obviously a ring homomorphism, and we call it an induced homomorphism [13] by .

Theorem 8 (see [13]). Let be an epimorphism of rings and let be an induced homomorphism of . If is an -invariant fuzzy ideal of , then

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 of a ring is called a fuzzy prime ideal [14] of if is a prime ideal of . A fuzzy set , defined as , is called a fuzzy nil radical [15] of .

Theorem 9 (see [15]). If is a fuzzy ideal of , then the fuzzy set is a fuzzy ideal of .

Lemma 10 (see [15]). If and are fuzzy ideals of , then .

Lemma 11. If and are fuzzy ideals of , then .

Proof. If is an element of , thenThis proves that .

Since is a fuzzy ideal of a polynomial ring by Theorem 5, the fuzzy set is the fuzzy nil radical of . The following theorem gives that the two fuzzy nil radicals have the same value.

Theorem 12. If is a fuzzy ideal of , then

Proof. Let be any element of . Then, by Theorem 5, we have . Since is a fuzzy ideal of , we obtainThis proves that .

Theorem 13. If and are fuzzy ideals of , then

Proof. If and are fuzzy ideals of , then and are fuzzy ideals of by Theorem 5. It follows from Theorems 12 and 7(i) and Lemma 10 thatproving the theorem.

Theorem 14. If and are fuzzy ideals of , then

Proof. Since and are fuzzy ideals of by Theorem 5, we obtainThis proves that .

Theorem 15. Let be a fuzzy ideal of and let be a homomorphism of rings. If is the induced homomorphism of , that is, , then

Proof. Given a polynomial , we haveThis proves that .

Proposition 16 (see [15]). Let be a ring epimorphism from onto , and let be a fuzzy ideal of . If is constant on , then .

Theorem 17. Let be a homomorphism of rings and let be the induced homomorphism of . If a fuzzy ideal of is constant on , then the fuzzy polynomial ideal is constant on .

Proof. Let for all and let be any element of . Then . It follows that for all . Hence for all ; that is, for all . This shows that , proving the theorem.

Corollary 18. Let be an epimorphism of rings and let be the induced homomorphism of . If an -invariant fuzzy ideal of is constant on , then

Proof. It follows from Proposition 16 and Theorem 8 that .

Theorem 19. Let be a fuzzy ideal of and let be its fuzzy polynomial ideal of . If is a fuzzy prime ideal of such that , then there exists a fuzzy prime ideal of such that and .

Proof. Since is a fuzzy prime ideal of , is a prime ideal of . If we define , then it is easy to show that is a prime ideal of . Define a fuzzy subset byThen, by routine calculations, we show that is a fuzzy ideal of satisfying . We claim that . Given , if , then . If , then . Since is a prime ideal of , is a prime ideal of . This shows that is a fuzzy prime ideal of , proving the theorem.

Definition 20 (see [15]). Let be a fuzzy ideal of . The fuzzy ideal defined byis called the prime fuzzy radical of .

Theorem 21. Let be a fuzzy ideal of and let be its fuzzy polynomial ideal of . Then

Proof. By Theorem 6, is a fuzzy prime ideal of with if and only if is a fuzzy prime ideal of with . It follows from Theorem 7(i) thatproving the theorem.

Notation 2. Let be a fuzzy ideal of and let be its fuzzy polynomial ideal of . We denote byand by

Theorem 22. Let be a fuzzy ideal of and let be its fuzzy polynomial ideal of . Then a map defined by is one-one.

Proof. If such that , then . It follows that for all , and hence for all , proving that . Hence is one-one.

Corollary 23. Let be a fuzzy ideal of and let be its fuzzy polynomial ideal of . If the map defined in Theorem 22 is an onto map, then

Proof. If is any element of , then there exists such that with . Thus . This shows that the reverse inclusion in Theorem 21 holds.

Example 24. Let be set of all integers. LetThen is a fuzzy prime ideal of , since is a prime ideal of , and its induced polynomial ideal is By Theorem 6, the fuzzy polynomial ideal induced by is a fuzzy prime ideal of . Hence .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

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