#### Abstract

In this work, we give a characterization of generalizations of prime and primary fuzzy ideals by introducing 2-absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals and establish relations between 2-absorbing (primary) fuzzy ideals and 2-absorbing (primary) ideals. Furthermore, we give some fundamental results concerning these notions.

#### 1. 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 2-absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals have not been studied yet. In this paper, we introduce the 2-absorbing 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 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 2-absorbing 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 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 , 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 directed collection of 2-absorbing primary fuzzy ideals of is 2-absorbing primary fuzzy ideal.

#### 2. Preliminaries

We assume throughout that all rings are commutative with . Unless stated otherwise 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).(ii). Let be any fuzzy ideal of ; , and let 0 be the additive identity of . Then it is easy to verify the following:(i), and , where and .(ii)If , then , iff , and .

*Definition 2 (see [7]). *Let be any fuzzy ideal of . The ideals , () are called level ideals of .

*Definition 3 (see [3]). *A fuzzy ideal of is called prime fuzzy ideal if for any two fuzzy points of , implies either or .

*Definition 4 (see [6]). *Let be a fuzzy ideal of . Then , called the radical of , is defined by .

*Definition 5 (see [6]). *A fuzzy ideal of is called primary fuzzy ideal if for , implies for some positive integer .

Theorem 6 (see [6]). *Let be fuzzy ideal of a ring . Then is a fuzzy ideal of .*

*Definition 7 (see [3]). *Let be a ring. Then a nonconstant fuzzy ideal is said to be weakly completely prime fuzzy ideal iff for , .

Theorem 8 (see [8]). *If and are two fuzzy ideals of , then .*

Theorem 9 (see [8]). *Let be a ring homomorphism and let be a fuzzy ideal of such that is constant on and let be a fuzzy ideal of . Then, *(i)*,*(ii)*.*

*Definition 10 (see [4]). *A nonzero proper ideal of a commutative ring with is called a 2-absorbing ideal if whenever with , then either or or .

*Definition 11 (see [5]). *A proper ideal of is called a 2-absorbing primary ideal of if whenever with , then either or or .

Theorem 12 (see [5]). *If is a 2-absorbing primary ideal of , then is a 2-absorbing ideal of .*

*Definition 13 (see [9]). *An element is called a 2-absorbing element if for any , implies either or or .

#### 3. 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 of , implies that either or or .

Theorem 15. *Every prime fuzzy ideal of is a 2-absorbing fuzzy ideal.*

*Proof. *The proof is straightforward.

Lemma 16. *Let be a fuzzy ideal and . If is a 2-absorbing fuzzy ideal, then is a 2-absorbing ideal of .*

*Proof. *Let be a 2-absorbing fuzzy ideal and . If for any , then . Thus, and . Since is a 2-absorbing fuzzy ideal, we have or or . Hence, or or . Therefore, is a 2-absorbing ideal of .

The following example shows that the converse of the lemma need not be true.

*Example 17. *Let , the ring of integers. Define the fuzzy ideal of by Then is , , in case , , , respectively. Thus, it is seen that is 2-absorbing ideal for all . Since so but and . Hence, is not 2-absorbing fuzzy ideal.

*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 a 2-absorbing element. If is the fuzzy ideal of defined byfor all , then is a 2-absorbing primary fuzzy ideal of .*

*Proof. *Assume that but and and for any . Then and and for all . In this case, and , and so , , and so . Since is a 2-absorbing primary ideal of , then we get and so . By our assumption we get and . Thus, or or , since is 2-absorbing element, which is a contradiction. Hence, is a 2-absorbing primary fuzzy ideal of .

Theorem 21. *Every 2-absorbing fuzzy ideal of is a 2-absorbing primary fuzzy ideal.*

*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 asBy Theorem 19 and Proposition 20 *μ* is a 2-absorbing primary fuzzy ideal. For , but so . Thus, is not a 2-absorbing fuzzy ideal.

Lemma 23. *Let be a fuzzy ideal and . If is a 2-absorbing primary fuzzy ideal, then is a 2-absorbing primary ideal of .*

*Proof. *If for any , then . Thus, and . Since is a 2-absorbing primary fuzzy ideal, we have or or . Hence, or or . Therefore, is a 2-absorbing primary ideal.

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 .*

*Proof. *Let be any fuzzy points of such that and . Since , we get . By the definition of , . Then there is a such that . It implies that . If , then, for all , . Since is a 2-absorbing primary fuzzy ideal, we conclude or . Thus, is a 2-absorbing fuzzy ideal.

*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 .

Theorem 26. *Let be -2-absorbing primary fuzzy ideals of for some 2-absorbing fuzzy ideal of . Then is a -2-absorbing primary fuzzy ideal of .*

*Proof. *Suppose that and . Then for some and for all . Since is a -2-absorbing primary fuzzy ideal, we have or . Thus, is a -2-absorbing primary ideal of .

In the following example, we show that if are 2-absorbing primary fuzzy ideals of a ring , then need not to be a 2-absorbing primary fuzzy ideal of .

*Example 27. *Let , the ring of integers. Define the fuzzy ideals and of byand byHere and are 2-absorbing primary fuzzy ideals of by Proposition 20. But it is not difficult to show that is not a 2-absorbing primary fuzzy ideal of . Since then but , , and . Moreover, by the definition of we conclude that but , , and . Hence, is not a 2-absorbing primary fuzzy ideal of .

Theorem 28. *Let be a fuzzy ideal of . If is a prime fuzzy ideal of , then is a 2-absorbing primary fuzzy ideal of .*

*Proof. *Assume that and for any and . Since and is commutative ring, we have . Thus, or since is a prime fuzzy ideal of . Hence, we conclude that is a 2-absorbing primary fuzzy ideal of .

Corollary 29. *If is a prime fuzzy ideal of , then is 2-absorbing primary fuzzy ideal of for any .*

*Proof. *Let be a prime fuzzy ideal and but for any . Since and is commutative ring, we conclude that . Hence, or since is prime fuzzy ideal of .

Theorem 30. *Let be a directed collection of 2-absorbing primary fuzzy ideals of . Then the fuzzy ideal is a 2-absorbing primary fuzzy ideal of .*

*Proof. *Suppose and for some fuzzy points of . Then there are some such that and for all . Since is 2-absorbing primary fuzzy ideal, we have or . Thus, or .

Theorem 31. *Let be a ring homomorphism. If is a 2-absorbing primary fuzzy ideal of , then is a 2-absorbing primary fuzzy ideal of .*

*Proof. *Assume that , where are any fuzzy points of . Then . Let , , and . Thus, we get that and . Since is a 2-absorbing primary fuzzy ideal, we conclude that or or . If , then ; hence, we conclude that .

If , then, for some , . Thus, we get that . By a similar way, it can be see that .

Theorem 32. *Let be a surjective ring homomorphism. If is a 2-absorbing primary fuzzy ideal of which is constant on , then is a 2-absorbing primary fuzzy ideal of .*

*Proof. *Suppose that , where are any fuzzy points of . Since is a surjective ring homomorphism, there exist such that , . Thus, because is constant on . Then we get . Since is a 2-absorbing primary fuzzy ideal, we conclude or or .

Thus, so or so .

By a similar way, it is easy to see that if .

#### 4. Weakly Completely 2-Absorbing Primary Fuzzy Ideals

*Definition 33. *(i) A nonconstant fuzzy ideal of is called a weakly completely 2-absorbing fuzzy ideal of if for all , or or .

(ii) A nonconstant fuzzy ideal of is called a weakly completely 2-absorbing primary fuzzy ideal of if for all , or or .

Theorem 34. *Every weakly completely 2-absorbing fuzzy ideal of 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.

*Example 35. *Let , the ring of integers. Define the fuzzy ideal of byAssume that for any . Thus, and so we get and . Since is primary ideal of , we get . By the definition of radical hence, or . Therefore, is a weakly completely 2-absorbing primary fuzzy ideal. But since , we conclude that is not a weakly completely 2-absorbing fuzzy ideal.

Proposition 36. *Every primary fuzzy ideal of is a weakly completely 2-absorbing primary fuzzy ideal.*

*Proof. *Let be a primary fuzzy ideal of . Assume that for any . Since is primary fuzzy ideal, we conclude . Since is a fuzzy ideal, we have or . So 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 .*

*Proof. *Assume that and for any . We show that or . Note that . Since is a weakly completely 2-absorbing primary fuzzy ideal of , we have or . Thus, or . So we conclude that is a 2-absorbing primary ideal of .

Conversely, assume that is a 2-absorbing primary ideal of for all . If for any , then there is a such that and . So and . Since is a 2-absorbing primary ideal of , we get that or . Hence, or . Therefore, a is weakly completely 2-absorbing primary fuzzy ideal of .

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 . 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.

*Definition 39. *Let be fuzzy ideal of . Then is called a -2-absorbing primary fuzzy ideal of if for all , implies that or or .

Proposition 40. *Every weakly completely 2-absorbing primary fuzzy ideal is a -2-absorbing primary fuzzy ideal.*

*Proof. *Assume that is a weakly completely 2-absorbing primary fuzzy ideal. If for any , then or or since is a weakly completely 2-absorbing primary fuzzy ideal. Hence, or or . We conclude that is a -2-absorbing primary fuzzy ideal.

Note that the following example shows that a -2-absorbing primary fuzzy ideal need not be a weakly completely 2-absorbing primary fuzzy ideal.

*Example 41. *Let , the ring of integers. Define the fuzzy ideal of byThen is a -2-absorbing primary fuzzy ideal. But sinceorthen 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.

Theorem 43. *Every -2-absorbing fuzzy ideal is a -2-absorbing primary fuzzy ideal.*

*Proof. *The proof is straightforward.

The following example shows that the converse of Theorem 43 is not true.

*Example 44. *Define the fuzzy ideal of by Then is a -2-absorbing primary fuzzy ideal but since and , we have that is not a -2-absorbing fuzzy ideal.

Theorem 45. *Let be a ring homomorphism. If is a weakly completely 2-absorbing primary fuzzy ideal of S, then is a weakly completely 2-absorbing primary fuzzy ideal of .*

*Proof. *Assume that for any . Then . Since is a weakly completely 2-absorbing primary fuzzy ideal of , we conclude that or . Thus, 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 , then is a weakly completely 2-absorbing primary fuzzy ideal of .*

*Proof. *Assume that for any Since is surjective ring homomorphism, , , and for some . Thus, . So, as is constant on , and . This means that . Since is a weakly completely 2-absorbing primary fuzzy ideal of , we have or . Hence, 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 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 , 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 is the corresponding weakly completely 2-absorbing primary fuzzy ideal of .*

*Remark 48. *Note that the following diagram shows the transition between definitions of fuzzy ideals:

#### 5. 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.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.