2-ABSORBING  -PRIMARY FUZZY IDEALS OF COMMUTATIVE RINGS

In this work, we define 2-absorbing  -primary fuzzy ideals which is the generalization of 2-absorbing fuzzy ideal and 2-absorbing primary fuzzy ideals. Furthermore, we give some fundamental results concerning these notions.


Introduction
The fundamental concept of fuzzy set was introduced by Zadeh [1] in 1965.In 1982, Liu introduced the notion of fuzzy ideal of a ring [2].Mukherjee and Sen have continued the study of fuzzy ideals by introducing the notion of prime fuzzy ideals [3].To the present day, fuzzy algebraic structures have been developed and many interesting results were obtained.
Prime ideals and primary ideals play a significant role in commutative ring theory.Because of this importance, the concept of 2-absorbing ideals, which is a generalization of prime ideals [4], and the concept of 2-absorbing primary ideals, which is a generalization of primary ideals [5], were introduced.While the prime fuzzy ideals and primary fuzzy ideals have been investigated [3,6], the concepts of 2absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals have not been studied yet.In this paper, we introduce the 2absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals and some generalizations of 2-absorbing primary fuzzy ideals and describe some properties of 2-absorbing primary fuzzy ideals.
Let  be a commutative ring with identity.Recall that a proper ideal  of  is called a 2-absorbing ideal if whenever , ,  ∈  and  ∈ , then either a ∈  or  ∈  or  ∈  [4] and a proper ideal  of  is called a 2-absorbing primary ideal if whenever , ,  ∈  and  ∈ , then either  ∈  or  ∈ √  or  ∈ √  [5].Based on these definitions, a nonconstant fuzzy ideal  of  is called a 2-absorbing fuzzy ideal of  if for any fuzzy points   ,   ,   of ,       ∈  implies that either     ∈  or     ∈  or     ∈  and a nonconstant fuzzy ideal  of  is said to be a 2-absorbing primary fuzzy ideal of  if for any fuzzy points   ,   ,   of ,       ∈  implies that either     ∈  or     ∈ √  or     ∈ √ .It is shown that if  is a 2-absorbing primary fuzzy ideal of , then √  is 2-absorbing fuzzy ideal.We introduce the notions of weakly completely 2-absorbing fuzzy ideal and weakly completely 2-absorbing primary fuzzy ideal, which is a weakened status of the 2-absorbing fuzzy ideals and 2absorbing primary fuzzy ideal, respectively.Then relationship between the 2-absorbing primary fuzzy ideals and weakly completely 2-absorbing primary fuzzy ideals is analyzed.Based on the definition of the level set, the transition of 2-absorbing primary ideals of  and 2-absorbing primary fuzzy ideals of  is examined.By a 2-absorbing primary ideal of , a 2-absorbing primary fuzzy ideal is established (Proposition 20).For a ring homomorphism  :  → , it is shown that  −1 () is a 2-absorbing primary fuzzy ideal of , where  is 2-absorbing primary fuzzy ideal of .If  is 2-absorbing primary fuzzy ideal of , which is constant on Ker , then it is proved that () is a 2-absorbing primary fuzzy ideal of .It is shown under what condition the intersection of the collection of 2-absorbing primary fuzzy ideals is 2-absorbing primary fuzzy ideal.It is shown that the intersection of two 2-absorbing primary fuzzy ideals need not be a 2-absorbing primary fuzzy ideal if this condition is not satisfied (Example 27).Also, it is proved that union of a 2 Mathematical Problems in Engineering directed collection of 2-absorbing primary fuzzy ideals of  is 2-absorbing primary fuzzy ideal.

Preliminaries
We assume throughout that all rings are commutative with 1 ̸ = 0. Unless stated otherwise  = [0, 1] stands for a complete lattice. denotes the ring of integers, () denotes the set of fuzzy sets of , and () denotes the set of fuzzy ideals of .For ,  ∈ (), we say  ⊆  if and only if () ≤ () for all  ∈ .When  ∈ , ,  ∈  we define   ∈ () as follows: and   is referred to as fuzzy point of .Also, for  ∈ () and  ∈ , define   as follows: Definition 1 (see [2]).A fuzzy subset  of a ring  is called a fuzzy ideal of  if for all ,  ∈  the following conditions are satisfied: (i) ( − ) ≥ () ∧ (), ∀,  ∈ .
Definition 3 (see [3]).A fuzzy ideal  of  is called prime fuzzy ideal if for any two fuzzy points   ,   of ,     ∈  implies either   ∈  or   ∈ .
Theorem 6 (see [6]).Let  be fuzzy ideal of a ring .Then √  is a fuzzy ideal of .
Theorem 9 (see [8]).Let  :  →  be a ring homomorphism and let  be a fuzzy ideal of  such that  is constant on Ker  and let  be a fuzzy ideal of .Then, Definition 10 (see [4]).A nonzero proper ideal  of a commutative ring  with 1 ̸ = 0 is called a 2-absorbing ideal if whenever , ,  ∈  with  ∈ , then either  ∈  or  ∈  or  ∈ .

2-Absorbing Primary Fuzzy Ideals
Definition 14.Let  be a nonconstant fuzzy ideal of .Then  is called a 2-absorbing fuzzy ideal of  if for any fuzzy points Theorem 15.Every prime fuzzy ideal of  is a 2-absorbing fuzzy ideal.
Proof.The proof is straightforward.
Example 17.Let  = , the ring of integers.Define the fuzzy ideal  of  by Definition 18.Let  be a nonconstant fuzzy ideal of .Then  is said to be a 2-absorbing primary fuzzy ideal of  if       ∈  implies that either     ∈  or     ∈ √  or     ∈ √  for any fuzzy points   ,   ,   .
Theorem 19.Every primary fuzzy ideal of  is a 2-absorbing primary fuzzy ideal of .
Proof.It is clear from the definition of primary fuzzy ideal.
Proposition 20.Let  be a 2-absorbing primary ideal of  and 1 ̸ =  ∈  a 2-absorbing element.If  is the fuzzy ideal of  defined by for all  ∈ , then  is a 2-absorbing primary fuzzy ideal of .
Proof.Proof.The proof is straightforward by the definition of the 2-absorbing fuzzy ideal.
The following example shows that the converse of Theorem 21 is not true.
Example 22.Let  ∈ () be defined as By Theorem 19 and Proposition 20  is a 2-absorbing primary fuzzy ideal.For 2 ∈ , 2 Note that if   is a 2-absorbing primary ideal of , then  need not be 2-absorbing primary fuzzy ideal of .In Example 17,  is not 2-absorbing fuzzy ideal and also it is not 2-absorbing primary fuzzy ideal by Theorem 21, although   is a 2-absorbing primary ideal of .
Proposition 24.If  is a 2-absorbing primary fuzzy ideal of , then √  is a 2-absorbing fuzzy ideal of .
Definition 25.Let  be a 2-absorbing primary fuzzy ideal of .Then  = √  is a 2-absorbing fuzzy ideal by Proposition 24.We say that  is a -2-absorbing primary fuzzy ideal of .
Proof.Suppose that       ∈  and     ∉ .Then     ∉   for some  ≥  ≥ 1 and       ∈   for all  ≥  ≥ 1.Since   is a -2-absorbing primary fuzzy ideal, we have Thus,  is a -2-absorbing primary ideal of .
In the following example, we show that if  1 ,  2 are 2absorbing primary fuzzy ideals of a ring , then  1 ∩  2 need not to be a 2-absorbing primary fuzzy ideal of .
Theorem 32.Let  :  →  be a surjective ring homomorphism.If  is a 2-absorbing primary fuzzy ideal of  which is constant on Ker , then () is a 2-absorbing primary fuzzy ideal of .

Theorem 34. Every weakly completely 2-absorbing fuzzy ideal of 𝑅 is a weakly completely 2-absorbing primary fuzzy ideal.
Proof.The proof is straightforward.
The following example shows that the converse of Theorem 34 is not necessarily true.
Proposition 36.Every primary fuzzy ideal of  is a weakly completely 2-absorbing primary fuzzy ideal.
Lemma 37. Let  be a fuzzy ideal of .Then  is a weakly completely 2-absorbing primary fuzzy ideal of  if and only if   is a 2-absorbing primary ideal of  for all  ∈ [0, (0)].
Theorem 38.If  is a weakly completely 2-absorbing primary fuzzy ideal of , then √  is a weakly completely 2-absorbing fuzzy ideal of .
Proof.If  is a weakly completely 2-absorbing primary fuzzy ideal, then by Lemma 37   is a 2-absorbing primary ideal of  for any  ∈ [0, (0)].By [5, Theorem 2.2], √   = √   is 2-absorbing ideal of .Then it is easy to see that from Definition 33 √   is a 2-absorbing ideal of  if and only if √  is a weakly completely 2-absorbing fuzzy ideal.
Example 41.Let  = , the ring of integers.Define the fuzzy ideal  of  by Then  is a -2-absorbing primary fuzzy ideal.But since then  is not a weakly completely 2-absorbing primary fuzzy ideal.
Corollary 42.Every weakly completely prime fuzzy ideal is a weakly completely 2-absorbing primary fuzzy ideal.
Proof.Since every weakly completely prime fuzzy ideal is primary fuzzy ideal, by Proposition 36 every weakly completely prime fuzzy ideal is a weakly completely 2-absorbing primary fuzzy ideal.
Proof.The proof is straightforward.
The following example shows that the converse of Theorem 43 is not true.
Theorem 45.Let  :  →  be a ring homomorphism.If  is a weakly completely 2-absorbing primary fuzzy ideal of S, then  −1 () is a weakly completely 2-absorbing primary fuzzy ideal of .
Theorem 46.Let  :  →  be a surjective ring homomorphism.If  is a weakly completely 2-absorbing primary fuzzy ideal of  which is constant on Ker , then () is a weakly completely 2-absorbing primary fuzzy ideal of .
We state the following corollary without proof.Its proof is a result of Theorems 45 and 46.
Corollary 47.Let  be a homomorphism of a ring  onto a ring .Then  induces a one-one inclusion preserving correspondence between the weakly completely 2-absorbing primary fuzzy ideals of  which is constant on Ker  and the weakly completely 2-absorbing primary fuzzy ideals of  in such a way that if  is a weakly completely 2-absorbing primary fuzzy ideal of  constant on Ker , then () is the corresponding weakly completely 2-absorbing primary fuzzy ideal of , and if  is a weakly completely 2-absorbing primary fuzzy ideal of , then  −1 () is the corresponding weakly completely 2-absorbing primary fuzzy ideal of .

Conclusion
This article investigates the weakly completely 2-absorbing primary fuzzy ideal and 2-absorbing primary fuzzy ideal as a generalization of primary fuzzy ideal in commutative rings.Also some characterizations of 2-absorbing primary fuzzy ideal are obtained.Moreover, we see that a 2-absorbing primary fuzzy ideal by a 2-absorbing primary ideal of a commutative ring is established, so the transition between the two structures can be analyzed.
Thus,  is not a 2-absorbing fuzzy ideal.Lemma 23.Let  be a fuzzy ideal and  ∈ [0, (0)].If  is a 2absorbing primary fuzzy ideal, then   is a 2-absorbing primary ideal of .