On 2-Absorbing Primary Fuzzy Ideals of Commutative Rings

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.