ON INTUITIONISTIC FUZZY PRIMARY IDEAL OF A RING

A BSTRACT . The purpose of this paper is to introduce and investigate primary ideal and P -primary ideal in the intuitionistic fuzzy environment and lay down the foundation for the primary decomposition theorem in the intuitionistic fuzzy setting. Also a suitable characterization of intuitionistic fuzzy P -primary ideal will be discussed.


INTRODUCTION
The decomposition of an ideal into primary ideals is a traditional pillar of ideal theory. It provides the algebraic foundation for decomposing an algebraic variety into its irreducible components. From another point of view primary decomposition provides a generalization of the factorization of an integer as a product of prime-powers. A prime ideal in a ring R is in some sense a generalization of a prime number. Also, primary ideal is some sort of generalization of prime ideal. An ideal Q in a ring R is called primary if Q = R and if xy ∈ Q ⇒ either x ∈ Q or y n ∈ Q for some n ∈ N.
In other words, Q is primary ideal ⇔ R/Q = 0 and every non-zero divisors in R/Q is nilpotent.
After the foundation of the theory fuzzy sets by Zadeh [17]. Mathematician started fuzzifying the algebraic concepts. Rosenfeld [13] was the first one to introduce the notion of fuzzy subgroup of a group. The concepts of fuzzy subring and ideal were introduced and studied by Liu in [9]. The notion like fuzzy (prime, primary, semi-prime, nil radical etc.) ideals were studied by Swamy at al. in [16] and Malik et al. in [10]. A detailed study of different algebraic structures in fuzzy setting can be found in [11].
One of the prominent generalizations of fuzzy sets theory is the theory of intuitionistic fuzzy sets introduced by Atanassov [1], [2] and [3]. Biswas introduced the notion of intuitionistic fuzzy subgroup of a group in [6]. The concepts of intuitionistic fuzzy subring and ideal were introduced and studied by Hur and others in [7]. The notion like intuitionistic fuzzy (prime, primary, semi-prime, nil etc.) ideals were studied in [4], [8], [12] and [15].
In this paper, we evaluate primary ideals and P -primary ideals in the intuitionistic fuzzy environment and laid down the foundation for the primary decomposition theorem in the intuitionistic fuzzy setting.
Further, if f : X → Y is a mapping and A, B be respectively IFS of X and Y . Then [4,7]) Let A ∈ IF S(R). Then A is called an intuitionistic fuzzy ideal (IF I) of ring R if for all x, y ∈ R, the followings are satisfied The set of all intuitionistic fuzzy ideals of R is denoted by IF I(R).
where as usual supremum and infimum of an empty set are taken to be 0 and 1 respectively. Note that AB ⊆ A * B.

Definition 2.10. ( [8]) Let
A be an IFI of a ring R. The intuitionistic nil radical of A is an IFS denoted by , and f is onto.

INTUITIONISTIC FUZZY PRIMARY IDEAL
In this section, we give a complete characterization of an intuitionistic fuzzy primary ideal of a commutative ring R with unity. Definition 3.1. Let Q be a non-constant IFI of a ring R. Then Q is said to be an intuitionistic fuzzy primary ideal of R, if for any two IFIs A, B of R such that AB ⊆ Q implies that either A ⊆ Q or B ⊆ √ Q.
From the definition, it is clear that every intuitionistic fuzzy prime ideal of R is an intuitionistic fuzzy primary ideal of R. But converse need not be true (see Example (3.8)).

Theorem 3.2.
Let Q be an IFI of a ring R. Then for any x (p,q) , y (t,s) ∈ IF P (R) the following are equivalent: (i) Q is an intuitionistic fuzzy primary ideal of R.
Thus Q is an intuitionistic fuzzy primary ideal of R.
The following theorem, which relates intuitionistic fuzzy primary ideal to primary ideal of the ring, will be needed in the proof of Theorem (3.7).

Theorem 3.3. Let A be an intuitionistic fuzzy primary ideal of R. Then
Hence A (α,β) is a primary ideal of R.
Remark 3.4. Let A be an intuitionistic fuzzy primary ideal of R. Then Remark 3.5. The converse of the Theorem (3.3) is not always true, see the following example: Clearly, A ∈ IF I(R). By some manipulation, we can see that, for all α, β ∈ (0, 1], A (α,β) is a primary ideal of R. But it can be easily checked that A is not an intuitionistic fuzzy primary ideal of R. For, if we take The following theorem characterized, intuitionistic fuzzy primary ideal completely.
Theorem 3.7. (a) Let J be a primary ideal of commutative ring R with unity 1 R and α, β ∈ (0, 1] such that for all x ∈ R. Then A is an intuitionistic fuzzy primary ideal of R. (b)Conversely, any intuitionistic fuzzy primary ideal can be obtained as in (a).
Proof. (a) Since J is a primary ideal of R, J = R, so A is non-constant intuitionistic fuzzy ideal of R. We show that A is an intuitionistic fuzzy primary ideal of R.
If µ A (ab) = 1 and ν A (ab) = 0, then ab ∈ J. Since b / ∈ J and J is a primary ideal of R, we have a n ∈ J, for some n ∈ N. Hence µ A (a n ) = 1 and ν A (a n ) = 0. Thus µ A (a n ) = 1 ≥ s and ν A (a n ) = 0 ≤ t implies that a n (s,t) ⊆ A, i.e., a (s,t) ⊆ √ A.
Hence A is an intuitionistic fuzzy primary ideal of R.
(b) Let A be an intuitionistic fuzzy primary ideal of R. We show that A is of the form for all x ∈ R, where α, β ∈ (0, 1] such that α + β ≤ 1.
Define IFSs C, D of R as C = χ A * , D(x) = A(0), ∀x ∈ R. It is easy to verify that C, D are IFIs of R. Since µ C (0) = 1 > µ A (0), ν C (0) = 0 < ν A (0) and µ D (a) = µ A (0) > µ A (a), ν D (a) = ν A (0) < ν A (a). So C is not a subset of A and D is not a subset of A and so C is not a subset of √ A and D is not a subset of √ A. Thus CD is not a subset of A, which is not true, since for all x, y ∈ R, we have µ A (xy) ≥ µ C (x) ∧ µ D (y) and ν A (xy) ≤ ν C (x) ∨ ν D (y). Thus µ A (0) = 1, ν A (0) = 0.

Claim (3) A has two values.
Since A * is a primary ideal of R, A * = R. Then there exists z ∈ R\A * . We will show that Take x = y, we have µ A (z) ≤ µ A (y) and ν A (z) ≥ ν A (y). Similarly, µ A (y) ≤ µ A (z) and ν A (y) ≥ ν A (z). Hence µ A (z) = µ A (y) and ν A (z) = ν A (y).
Hence, every intuitionistic fuzzy primary ideal of R, is of the form for all x ∈ R, where J = A * is a primary ideal of R.
Theorem 3.8. If J be a non-trivial ideal in a ring R, then χ J is an intuitionistic fuzzy primary ideal of R if, and only if, J is a primary ideal of R.
Theorem (3.8) is particularly useful in deciding whether, an intuitionistic fuzzy ideal, is primary or not. The following example illustrate this. Example 3.9. Let R = Z, be the ring of integers. Then where p is a prime integer and k > 1. Then, by Theorem (3.8), A is an intuitionistic fuzzy primary ideal of Z, since < p k > is a primary ideal of Z. Notice that A is not an intuitionistic fuzzy prime ideal of R.
In the following example, we show that, if Q 1 , Q 2 are two intuitionistic fuzzy primary ideals of a ring R, then Q 1 ∩ Q 2 need not be an intuitionistic fuzzy primary ideal of R. Example 3.10. Let R = Z, be the ring of integers. Take I = 2Z, J = 3Z. Clearly, I and J are primary (in fact prime) ideals in R. Define Q 1 = χ I , Q 2 = χ J . Then, by Theorem (3.8), Q 1 and Q 2 are intuitionistic fuzzy primary ideals of R. Also, Q 1 ∩ Q 2 = χ I∩J = χ 6Z , which is not an intuitioniatic fuzzy primary ideal of R, as 6Z is not a primary ideal in Z. Definition 3.11. An intuitionistic fuzzy primary ideal Q of ring R with √ Q = P , is called an intuitionistic fuzzy P -primary ideals of ring R. Theorem 3.12. Let Q 1 , Q 2 , ......, Q n be intuitionistic fuzzy P -primary ideals of ring R with P = √ Q i , ∀i = 1, 2, ..., n, an intitionistic fuzzy prime ideal of R. Then Q = n i=1 Q i is an intuitionistic fuzzy P -primary ideal of R.
In the next theorems we show that, both the image and inverse image of an intuitionistic fuzzy primary (P -primary) ideal under a ring epimorphism are again intuitionistic fuzzy primary (P -primary) ideal. Theorem 3.13. Let f be an ring epimorphism from R to R 1 . If A is an intuitionistic fuzzy primary ideal of R such that χ kerf ⊆ A, then f (A) is an intuitionistic fuzzy primary ideal of R 1 .
We show that f (A) is an intuitionistic fuzzy primary ideal of R 1 . Since A is an intuitionistic fuzzy primary ideal of R, so A is of the form for all x ∈ R, where α, β ∈ (0, 1] such that α + β ≤ 1 and J = A * is a primary ideal of R.
We first claim that, if A * is a primary ideal of R and For all a 1 , b 1 ∈ R 1 , since f is epimorphism, therefore there exists a, b ∈ R such that a 1 = f (a) and b 1 = f (b). Now, ab ∈ A * and A * is a primary ideal of R, so either a ∈ A * or b n ∈ A * , for some n ∈ N.
Thus f (A * ) is a primary ideal of R 1 . So by Theorem (3.7), for all y ∈ R 1 , Hence f (A) is an intuitionistic fuzzy primary ideal of R 1 .
Theorem 3.14. Let f be an ring epimorphism from R to R 1 . If A is an intuitionistic fuzzy P -primary ideal of R such that χ kerf ⊆ A, then f (A) is an intuitionistic fuzzy f (P )-primary ideal of R 1 .
Proof. This follows from Theorem (3.13) and Theorem (2.13)(ii) Theorem 3.15. Let f be a ring epimorphism from R to R 1 . If B is an intuitionistic fuzzy primary ideal of R 1 , then f −1 (B) is an intuitionistic fuzzy primary ideal of R.
Proof. Let B be an intuitionistic fuzzy primary ideal of R 1 . Then for all y ∈ R 1 and B * is a primary ideal of R 1 .
We first show that f −1 (B * ) is a primary ideal of R.
Hence f −1 (B) is an intuitionistic fuzzy primary ideal of R.
Theorem 3.16. Let f be a ring epimorphism from R to R 1 . If B is an intuitionistic fuzzy P -primary ideal of R 1 , then f −1 (B) is an intuitionistic fuzzy f −1 (P )-primary ideal of R.

INTUITIONISTIC FUZZY PRIMARY DECOMPOSITION
In this section, we study the decomposability of an intuitionistic fuzzy ideal in a Noetherian ring, in terms of intuitionistic fuzzy primary ideals, such that the set of their respective intuitionistic fuzzy radical ideals, are independent of the particular decomposition.
To begin this section, we first recall from [14] the definition of residual quotient (A : B) of an intuitionistic fuzzy ideal A by an intuitionistic fuzzy set B in a ring R.
For any IFP x (p,q) of the ring R, we use a streamlined notation (A : x (p,q) ) for (A : x (p,q) ), where x (p,q) = {C : C is an IFI of R such that x (p,q) ⊆ C}, be an IFI generated by x (p,q) . There is no difficulty in seeing that (A : x (p,q) ) is an IFI of R and A ⊆ (A : x (p,q) ).
Proof. Let x (p,q) ∈ IF P (R), Q be an intuitionistic fuzzy primary ideal of R such that P = √ Q.
(iii) Since Q ⊇ x (p,q) ∩ Q ⊇ x (p,q) Q, i.e., x (p,q) Q ⊆ Q. Therefore, by the properties of IF residual quotient, we have Q ⊆ (Q : x (p,q) ). Further, x (p,q) (Q : x (p,q) ) ⊆ Q. As Q is an intuitionistic fuzzy primary ideal of R and x (p,q) / ∈ √ Q implies that (Q : x (p,q) ) ⊆ Q. Hence (Q : x (p,q) ) = Q.  x (p,q) ).  (1) all intuitionistic fuzzy primary ideal Q k = P for j = k, then we may achieve (1) of definition (4.5) by replacing Q j and Q k by Q = Q j ∩ Q k which is an intuitionistic fuzzy P -primary ideal of R, by Theorem (3.12). Repeating this process, we get will arrive at an intuitionistic fuzzy primary decomposition in which all √ Q i are distinct. If n j =i=1 Q j ⊆ Q i , we may simply omit Q i . Repeating this process, we will achieve (2) of definition (4.5).
Lemma 4.7. Let A 1 , A 2 , ...., A n be IFIs of ring R and let P be an intuitionistic fuzzy prime ideal of R. Then But, since P is an intuitionistic fuzzy prime ideal and A 1 A 2 ....A n ⊆ P , then A i ⊆ P for some i.
(2) If P = n i=1 A i , then P ⊆ A i for some i, and from part (1), A i ⊆ P for some i. Hence P = A i , for some i. Definition 4.8. An intuitionistic fuzzy prime ideal P in a ring R is called an intuitionistic fuzzy associated prime ideal of an IFI A, if P = (A : x (p,q) ) for some x (p,q) ∈ IF P (R). Moreover, for an IFI A of a ring R. We define IF − ASS(A) to be the set of all intuitionistic fuzzy prime ideals associated with the IFI A, i.e., Q i , 1 ≤ i ≤ n be the minimal intuitionistic fuzzy primary decomposition of A. Consider any x (p,q) ∈ IF P (R), we have Hence (A : x (p,q) ) = n i=1 (Q i : x (p,q) ). Also, by Theorem (4.2), if x (p,q) ∈ Q j then (Q j : x (p,q) ) = χ R and if, x (p,q) / ∈ Q j , then (Q j : x (p,q) ) = P j , be an intuitionistic fuzzy prime ideal of R. So Now, suppose that P ∈ IF − ASS(A), then P = (A : x (p,q) ) be an intuitionistic fuzzy prime ideal of R, for some x (p,q) ∈ IF P (R). Since (A : x (p,q) ) = x (p,q) / ∈Qj P j , then by Lemma (4.7)(2) we have (A : x (p,q) ) = P j for some j. So, P ∈ {P i , i = 1, 2, ..., n}. Therefore, IF − ASS(A) ⊆ {P i , i = 1, 2, ..., n}.
Conversely, as the decomposition is minimal so n j =i=1 Q j is not a subset of Q i . Then for each i ∈ {1, 2, ..., n}, there exists (x i ) (pi,qi) ∈ n j =i=1 Q j and (x i ) (pi,qi) / ∈ Q i , we have (Since all other's (Q j : (x j ) (pj ,qj ) ) = χ R , for j = i by Theorem (4.2)).
Define IFSs A i on R as follows: where α i , β i ∈ (0, 1) such that α i + β i ≤ 1, for 1 ≤ i ≤ k and α k+1 = α 1 , β k+1 = β 1 and R i = Z p n 1 × ....... × Z p n k k is a primary ideal of R. Clearly, A i are intuitionistic fuzzy primary ideal of R. It can be easily checked that A = ∩ n i=1 A i is an intuitionistic fuzzy primary decomposition of A.
Example 4.11. Consider R = ∞ i=1 Z 2 , a direct product of infinitely many copies of the field Z 2 = {0,1} be a boolean ring. Then R is a ring, which is not a Noetherian ring, as the strictly ascending chain of ideals 0 ⊂ Z 2 × 0 ⊂ Z 2 × Z 2 × 0 ⊂ ...... is not stationary. For every t i , s i ∈ [0, 1) such that t i + s i ≤ 1, define A i ∈ IF S(R) as for all x ∈ R. Then by Theorem (2.9), A i is an intuitionistic fuzzy prime ideal and hence primary ideal of R.
Consider the IFI A of R defined by A(x) = (0, 1), ∀x ∈ R. Then A has no intuitionistic fuzzy primary decomposition in R, i.e., A = n i=1 A i , for any n ∈ N.

CONCLUSION
In this paper, we explored the fundamental ideas of intuitionistic fuzzy primary and P -primary ideal of a commutative ring R. We proved that an intuitionistic fuzzy primary ideal is a two valued intuitionistic fuzzy set with base set as a primary ideal (the base set of an intuitionistic fuzzy ideal A is defined as the set {x ∈ R|µ A (x) = µ A (0); ν A (x) = ν A (0)} and vice versa. We also investigated the behaviour of intuitionistic fuzzy primary ideal under ring homomorphism. The structure of intuitionistic fuzzy P -primary ideal of a commutative R has been fully explored. Many properties of intuitionistic fuzzy primary ideals have been studied in terms of residual quotients. We have also laid down the foundation of the most important property in ring theory: decomposition of an ideal in terms of primary ideals in the intuitionistic fuzzy environment.