On the intuitionistic fuzzy polynomial ideals of a ring

: In this paper we introduce the notion of intuitionistic fuzzy polynomial ideal A x of a polynomial ring R [ x ] induced by an intuitionistic fuzzy ideal A of a ring R , and obtain an iso-morphism theorem of a ring of intuitionistic fuzzy cosets of A x . It is 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, we show that if A x is an intuitionistic fuzzy maximal ideal of R [ x ] , then A is an intuitionistic fuzzy maximal ideal of R but converse is not true.


Introduction
One of the remarkable generalizations of the fuzzy sets 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 [6] and [7] defined and studied intuitionistic fuzzy subrings and ideals of a ring. With a different approach, Mukerjee and Basnet in [4] also studied intuitionistic fuzzy subrings of a ring. Jun and others 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 [14]. Meena and Thomas in [13] 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 [12]. In this paper, we introduce the notion of intuitionistic fuzzy polynomial ideal of a ring and study some of 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 2.1. ( [3,4]) 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 Definition 2.2. ( [8]) Let R and S be any sets and let f : R → S be a function.
If A is any f -invariant IFS of R, then f −1 (f (A)) = A.
The following results are easy to prove: Let R be a commutative ring with identity and let R[x] be the ring of polynomials where x is an indeterminate.
obviously a ring homomorphism, and we call f x an induced homomorphism by f .

Intuitionistic fuzzy polynomial ideals
In this section, we introduce the notion of intuitionistic fuzzy polynomial ideal of a ring and study their properties. The set of all real numbers is denoted by R.
Proof. Since A = (µ A , ν A ) be an IFI of a ring R. for any a i , b i ∈ R(i = 1, 2, . . . , n). By Definition (2.1), we have . Then by Lemma (3.1), we have Thus, µ Ax (f (x) − g(x)) ≥ min{µ Ax (f (x)), µ Ax (g(x))}. Similarly, we can show that Proposition 3.7. Let A be an IFI of the ring R. Then the set Proof. Let f (x), g(x) be any two element of S, then On the other hand, Remark 3.8. 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, Proof. It follows that Theorem 3.10. If A and B are two IFIs of a ring R, then .
Thus, we get µ AxBx (f (x)) ≤ µ (AB)x (f (x)). Similarly, we can show that Theorem 3.11. Let f : R → R be a homomorphism from R onto R . If A and B are IFIs of R , then Proof. Let x ∈ R be any element.
Theorem 3.14. Let f : R → R be a homomorphism from R onto R and let f x be an induced Proof. For any polynomial s( Similarly, we can show that ν fx(Ax) (s(x)) = ν Ax (h j (x)). Now, for i = 1, 2, . . . , m. As A is f -invariant, we have Proof. Straightforward result.
Lemma 3.17. Let A be an IFI of a ring R and let A x be an intuitionistic fuzzy polynomial Consider h(x) ∈ R[x] be any element, then we have Similarly, we can show that In a same way, we can show that f (x) + A x ⊆ g(x) + A x . Which complete the proof.
Theorem 3.18. Let A be an IFI of a ring R and let A x be an IFPI of R[x]. Then Proof. Define an map γ : Then it is easy to see that the map γ is an epimorphism of rings with Kerγ, where The result follows by first theorem of homomorphism of rings.

Prime and maximal intuitionistic fuzzy polynomial ideals
In this section, we study some properties of the prime and maximal intuitionistic fuzzy polynomial ideals.
Definition 4.1. An intuitionistic fuzzy ideal P of a ring R, not necessary constant, is said to be an intuitionistic fuzzy prime ideal, if for any IFIs A and B of R the condition AB ⊆ P implies that either A ⊆ P or B ⊆ P .

Proposition 4.2. Let
A is an intuitionistic fuzzy prime ideal of a ring R, then A * is a prime ideal of R.
is an intuitionistic fuzzy prime ideal of R.
. Let i be the first smallest non-negative integer such that µ A (a i ) = µ A (0) and ν A (a i ) = ν A (0) and let j be the first smallest non-negative integer such that Conversely, assume that A x is an intuitionistic fuzzy prime ideal of , either (ax) ∈ (A x ) * or (bx) ∈ (A x ) * , which shows that either a ∈ A * or b ∈ A * . This proves that A is an intuitionistic fuzzy prime ideal of R. Proof. Firstly, assume that B is an intuitionistic fuzzy prime ideal of R . Then B * is a prime ideal of R . Clearly, f −1 (B) is an IFI of R. We claim that (f −1 (B)) * is a prime ideal of R. Let a, b ∈ R be any element such that ab ∈ (f −1 (B)) * . Then µ f −1 (B) (ab) = µ f −1 (B) (0) and Proof. Firstly, assume that A is an intuitionistic fuzzy prime ideal of R. Then A * is a prime ideal of R. Let x, y ∈ R such that xy ∈ f (A * ). Since f is onto, there exists c ∈ A * such that f (c) = xy and there exists a, b ∈ R such that f (a) = x, f (b) = y. Since f (ab) = f (a)f (b) = xy = f (c).
As A is f -invariant, therefore, µ A (ab) = µ A (c) = µ A (0) and ν A (ab) = ν A (c) = ν A (0). Thus ab ∈ A * . Since A * is a prime ideal of R, either a ∈ A * or b ∈ A * , which shows that either Conversely assume that f (A * ) is a prime ideal of R and let a, b ∈ R such that ab ∈ A * .
, either a ∈ A * or b ∈ A * . Hence A * is a prime ideal of R and hence A is an intuitionistic fuzzy prime ideal of R.