Radical Structures of Fuzzy Polynomial Ideals in a Ring

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 α x 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 α x . It was shown that a fuzzy ideal α of a ring is fuzzy prime if and only if α x is a fuzzy prime ideal of R[x]. Moreover, we showed that if α x 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. 2. Preliminaries


Introduction
Zadeh [1] introduced the notion of a fuzzy subset of a set as a function from 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: -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.

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 : → [0, 1] satisfying the following axioms: Definition 2 (see [2]). Let : → be a homomorphism of rings and let be a fuzzy subset of . We define a fuzzy subset −1 of by −1 ( ) fl ( ( )) 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 by for all ∈ . The intersection of and , symbolized by ∩ , is a fuzzy subset of , defined by for all ∈ .
Theorem 5 (see [13]). Let : → [0, 1] be a fuzzy ideal of a ring and let ( ) The fuzzy ideal discussed in Theorem 5 is called the fuzzy polynomial ideal [13] of [ ] induced by a fuzzy ideal .
Theorem 6 (see [13]  Theorem 7 (see [13]). If and are fuzzy ideals of a ring , then Let : → be a homomorphism of rings. A map : is obviously a ring homomorphism, and we call it an induced homomorphism [13] by .

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 : → [0, 1] of a ring is called a fuzzy prime ideal [14] of if * is a prime ideal of . A fuzzy set √ : → [0, 1], defined as √ ( ) := ⋁{ ( ) | > 0}, is called a fuzzy nil radical [15] of . Theorem 9 (see [15]). If : → [0, 1] is a fuzzy ideal of , then the fuzzy set √ is a fuzzy ideal of .

Theorem 13. If and are fuzzy ideals of , then
Discrete Dynamics in Nature and Society proving the theorem.

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 ( ( )) = ( ) . (20) Proof. If is any element of FPI( ), then there exists ∈ FPI( ) such that = ( ) = with ⊆ . Thus ( ( )) = ( ). This shows that the reverse inclusion in Theorem 21 holds.
Example 24. Let Z be set of all integers. Let Then is a fuzzy prime ideal of Z, since * = 2Z is a prime ideal of Z, and its induced polynomial ideal is By Theorem 6, the fuzzy polynomial ideal induced by is a fuzzy prime ideal of Z[ ]. Hence ( ( )) = = ( ).