Radical structures of intuitionistic fuzzy polynomial ideals of a ring

: In this paper we investigate the radical structure of an intuitionistic fuzzy polynomial ideal A x induced by an intuitionistic fuzzy ideal A of a ring and study its properties. Given an intuitionistic fuzzy ideal B of a ring R (cid:48) and a homomorphism f : R → R (cid:48) , we show that if f x : R [ x ] → R (cid:48) [ x ] is the induced homomorphism of f , that is, f x ( (cid:80) ni =0 a i x i ) = (cid:80) ni =0 ( f ( a i )) x i , then f − 1 x [( √ B ) x ] = ( (cid:112) f − 1 ( B )) x .


Introduction
One of the remarkable generalizations of the fuzzy sets [14] is the intuitionistic fuzzy sets which was introduced by Atanassov [1,2]. Biswas was the first one to introduce the intuitionistic fuzzification of the algebraic structure and developed the concept of intuitionistic fuzzy subgroup of a group in [5]. Later on Hur and others in [7] and [6] defined and studied intuitionistic fuzzy subrings and ideals of a ring. With a different approach Banerjee and Basnet in [4] also studied intuitionistic fuzzy subrings and ideals of a ring. Jun and other in [8] introduced and study the notion of intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings. Translate of intuitionistic fuzzy subring and ideal was studied by Sharma in [12]. Meena and Thomas in [11] studied the concept of intuitionistic fuzzy subring to lattice setting and introduced the notion of intuitionistic L-fuzzy subring. The concept of Characteristic intuitionistic fuzzy subrings of an intuitionistic fuzzy ring was introduced by Meena in [10]. The present authors [13] introduced the notion of an intuitionistic fuzzy polynomial ideal A x of a polynomial ring R[x] induced by an intuitionistic fuzzy ideal A of a ring R and obtained an isomorphism theorem of a ring of an intuitionistic fuzzy cosets of A x . It was shown that an intuitionistic fuzzy ideal A of a ring is an intuitionistic fuzzy prime if and only if A x is an intuitionistic fuzzy prime ideal of R[x]. Moreover, it was shown that if A x is an intuitionistic fuzzy maximal ideal of R[x], then A is an intuitionistic fuzzy ideal of R .
In this paper, we investigate the radical structure of intuitionistic fuzzy polynomial ideal induced by an intuitionistic fuzzy ideal of a ring and study its properties.

Preliminaries
Definition 2.1. ( [1]) Let X be a non-empty fixed set. An intuitionistic fuzzy set (IFS) A in X is an object having the form ) and the degree of non-membership (namely ν A (x)) of each element x ∈ X to the set A respectively and [3,4,6,11]) Let R be a ring. An IFS A = (µ A , ν A ) of R is said to be an intuitionistic fuzzy ideal (IFI) of R if (d) If A and B are two IFIs of the ring R, then sum A + B and the product AB are defined as: Definition 2.7. ( [8]) Let R and S be any sets and let f : is obviously a ring homomorphism, and we call f x an induced homomorphism by f .

The intuitionistic fuzzy ideal A x is called the intuitionistic fuzzy polynomial ideal of R[x]
induced by an intuitionistic fuzzy ideal A of R.
be an induced homomorphism of f . If A is an IFI of the ring R and A x be its the intuitionistic Proposition 2.11. ( [13]) Let A be an IFI of the ring R. Then the set Remark 2.12. ( [11]) Let A be an IFS of a ring R. We denote a level cut set A * by It is proved in [11] that if A is an IFI of ring R, then A * is an ideal of ring R. Note that if A is an IFI of a ring R, Theorem 2.14. ( [13]) If A and B are two IFIs of a ring R, then

Radical of the intuitionistic fuzzy polynomial induced by an intuitionistic fuzzy ideal
In this section, we study some relations between the radical of the intuitionistic fuzzy polynomial ideal R[x] induced by an intuitionistic fuzzy ideal of a ring R and the radical of an intuitionistic fuzzy ideal of the ring.
be an intuitionistic fuzzy ideal of R. Then the intuitionistic fuzzy nil radical of A is defined to be an IFS Proof. Straightforward. Proof. Let x, y ∈ R. Then Let m and n be any positive integers. Since R is commutative, we know that each term in the binomial expansion of (x + y) m+n contains either x m or y n as a factor. Hence there exist r, t ∈ R such that (x + y) m+n = rx m + ty n . Thus Thus, From (3.5) and (3.6) we get µ √ A is an intuitionistic fuzzy ideal of R.
Theorem 3.4. If A and B are intuitionistic fuzzy ideals of R, then For another inclusion, let x ∈ R be any element. Then . Now, let m and n be any positive integers. Then, This completes the proof of (i). (ii) The proof follows similar to the proof of part (i). (iii) The proof follows from the definition of sum of IFIs. (iv) The proof follows from the definition of product of IFIs.
If A is an intuitionistic fuzzy ideal of a ring R, then A x is an intuitionsitic fuzzy ideal of a polynomial ring R[x] by Theorem 2.9, the IFS √ A x is the intuitionistic fuzzy nil radical of A x . The following theorem gives that the two intuitionistic fuzzy nil radicals have the same value.
Since √ A x is an intuitionistic fuzzy ideal of R[x], we obtain Similarly, we can show that ν ( Theorem 3.6. If A and B are intuitionistic fuzzy ideals of R, then Proof. Let A be an intuitionistic fuzzy ideal of R, then A x and B x are intuitionistic fuzzy polynomial ideals of R[x] by Theorem 2.9. For (i), we have For (ii), we have For (iii), we have For (iv), we have This completes the proof.
Theorem 3.7. Let B be an intuitionistic fuzzy ideal of R and let f : Similarly, we can show that µ ( ).
Proposition 3.8. Let f : R → R be an epimorphism from R onto R and let A be an intuitionistic fuzzy ideal of R, then f ( Proof. Clearly, f (A) and f ( √ A) are intuitionistic fuzzy ideals of R . If y ∈ R and f (x) = y for some x ∈ R, then f (x n ) = y n , for all n = 1, 2, . . . , then Similarly, we can show that ν f ( Further, if A is constant on Kerf and x 0 ∈ f −1 (y) is a fixed element, then by Proposition Similarly, we can show that µ f ( Theorem 3.9. Let f : R → R be a homomorphism from R onto R and let f x be the induced homomorphism of f . If an intuitionistic fuzzy ideal A of R is constant on Kerf , then the intuitionistic fuzzy polynomial ideal A x is constant on Kerf x . Proof. It follows from Proposition 3.8 and Theorem 3.5 that Definition 3.11. Let A be an intuitionistic fuzzy ideal of a ring R. Then the intuitionistic fuzzy ideal P (A) defined by where B is an intuitionistic fuzzy prime ideal of R } is called an intuitionistic fuzzy prime radical of A.
It follows from Theorem 2.14 (i) that This proves the theorem. Proof. Let B, C ∈ IF P I(A) such that φ(B) = φ(C), then B x = C x . It follows that (B x )(r) = (C x )(r), for all r ∈ R, and hence B(r) = C(r) for all r ∈ R, proving that B = C. Hence φ is one-one.
Corollary 3.15. Let A be an intuitionistic fuzzy ideal of a ring R and let A x be its intuitionistic fuzzy polynomial ideal of R[x]. If the map φ defined in Theorem 3.14 is one-one map, then (P (A)) x = P (A x ).
Proof. If D is any element of IF P I(A x ), then there exists C ∈ IF P I(A) with A ⊆ C such that C x = φ(C) = D. Thus (P (A)) x = P (A x ).
Example 3.16. Let Z be the set of all integers. Define an IFS A on Z by Then A is an intuitionistic fuzzy prime ideal of Z, for A * = 2Z is a prime ideal of Z, and its induced polynomial ideal A x is given by By Theorem 2.16, the intuitionistic fuzzy polynomial ideal A x induced by A is an intuitionistic fuzzy polynomial ideal of Z[x]. Hence (P (A)) x = A x = P (A x ).