Abstract
We investigate -prime and -primary elements in a compactly generated multiplicative lattice . By a counterexample, it is shown that a -primary element in need not be primary. Some characterizations of -primary and -prime elements in are obtained. Finally, some results for almost prime and almost primary elements in with characterizations are obtained.
1. Introduction
A multiplicative lattice is a complete lattice provided with commutative, associative, and join distributive multiplication in which the largest element 1 acts as a multiplicative identity. An element is called compact if, for , implies the existence of a finite number of elements in such that . The set of compact elements of will be denoted by . A multiplicative lattice is said to be compactly generated if every element of it is a join of compact elements. Throughout this paper denotes a compactly generated multiplicative lattice with 1 compact in which every finite product of compact elements is compact.
An element is said to be proper if . A proper element is called a prime element if implies or , where , and is called a primary element if implies or for some , where . A proper element is said to be weakly prime if implies either or , where , and is called weakly primary if implies or for some , where . For , . The radical of is denoted by and is defined as . An element is called semiprimary if is a prime element and is called semiprime if . An element is called join irreducible if implies or . A proper element is said to be a maximal element if for any other proper element . An element is said to be nilpotent if for some . An element is called a zero divisor if for some and is called an idempotent if . A multiplicative lattice is said to be a domain if it is without zero divisors and is said to be quasi-local if it contains a unique maximal element. A quasi-local multiplicative lattice with maximal element is denoted by . An element is called meet principal if for all . An element is called join principal if for all . An element is called weak meet principal if for all . An element is called weak join principal if for all . An element is called principal if is both meet principal and join principal. An element is called weak principal if is both weak meet principal and weak join principal. A multiplicative lattice is a Noether lattice if it is modular, principally generated (every element is a join of principal elements) and satisfies the ascending chain condition. A Noether lattice is local if it contains precisely one maximal prime. In a Noether lattice , an element is said to satisfy restricted cancellation law if, for any , implies [1]. The reader is referred to [2] for general background and terminology.
2. -Prime and -Primary Elements in
The study of -prime and -primary ideals for commutative rings is carried out by Anderson and Bataineh [3], Bataineh and Kuhail [4], and Yousefian Darani [5]. We extend these concepts to compactly generated multiplicative lattices. We introduce the notion of -prime, -primary, and -primary elements in .
Definition 1. Let be a function. A proper element is said to be -prime if, for , and implies either or .
Definition 2. Let be a function. A proper element is said to be -primary if, for , and implies either or .
Definition 3. Let be a function. A proper element is said to be -primary (or briefly -primary) if, for , and implies either or .
Similarly, -primary elements in are defined by following settings in the above definition for -primary element. For any proper element ,(i) and then is called weakly primary element;(ii) and then is called almost primary element;(iii) and then is called -almost primary element.
Thus an almost primary element in is a particular case of a -primary element in . Similarly an almost prime element in is a particular case of a -prime element in . Therefore their definitions are as follows.
Definition 4. A proper element is said to be almost prime if, for , and implies either or .
Definition 5. A proper element is said to be almost primary if, for , and implies either or .
The reader can verify the following statements. (1) for all . (2) Every -prime element in is -primary.
(Its converse holds for semiprime elements.) (3) Every prime element in is -prime. (4) Every prime element in is -primary. (5) Every primary element in is -primary.
The following example (take for convenience) shows that ① a -primary element in need not be -prime; ② a -primary element in need not be prime.
Example 6. Consider the lattice of ideals of the ring . Then the only ideals of are the principal ideals (0), (2), (3), (4), (6), (8), (12), (1). Clearly is a compactly generated multiplicative lattice. Its lattice structure is as shown in Figure 1.
From the multiplication table (see Figure 1), we see that element is almost primary, while is not almost prime because but and . Also, is not prime.
Theorem 7. If a proper element is -primary such that , then is a -prime element in .
Proof. Let . Assume that and but for some . Then there exist such that . If , then by hypothesis , a contradiction. So . Since is -primary and for all , we have and hence . This shows that is a -prime element in .
Definition 8. Given two functions , we define if for each .
Theorem 9. Let be functions such that . Then every proper -primary element in is -primary.
Proof. Let a proper element be -primary. Suppose , for some . Then as we have . Since is -primary, it follows that either or and hence is -primary.
Theorem 10. Let be a proper element. Then is primary implies is weakly primary implies is -primary implies is -almost primary implies is almost primary.
Proof. Obviously is primary implies is weakly primary.
Assume that is weakly primary but not -primary. Then there exist such that , , , and . As is compactly generated, there exist such that , , , and. Let and be any two compact elements of . Then such that , , and. Since is weakly primary, it follows that . So , which implies , a contradiction to . Hence is -primary.
Next we show that is -primary implies is -almost primary . Assume is -primary and . Let , for some . Then , . Since is -primary, it follows that either or . Hence, is -almost primary .
The last implication is obvious (from ).
From this theorem we get the following characterization of a -primary element in .
Corollary 11. Let be a proper element. Then is -primary if and only if is -almost primary for every .
Proof. Suppose is -almost primary for every . Let and for some . Then and for some . Since is -almost primary, we have or . Hence is -primary. The converse follows from Theorem 10.
Clearly every primary element in is -primary. But the converse is not true as shown in the following example by taking .
Example 12. Consider the lattice of ideals of the ring . Then the only ideals of are the principal ideals (0), (2), (3), (5), (6), (10), (15), (1). Clearly is a compactly generated multiplicative lattice. Here the element is almost primary but not primary.
In the following successive three theorems, we show conditions under which a -primary element in is primary.
Now we have a characterization of a -primary element in .
Theorem 13. Let be a Noether lattice. Let be a non-nilpotent proper element satisfying the restricted cancellation law. Then is -primary for some if and only if is primary.
Proof. Suppose is a primary element. Then obviously is -primary for every and hence for . Conversely, let be -primary for some . Then, by Theorem 9, is -primary (almost primary). Let for some . If , then as is -primary we have or . If , consider . If , then as is -primary we have or for some and hence or . So assume . Then and . By Lemma 1.11 of [1] we get which shows that is primary.
Definition 14. A proper element is said to be 2-potent prime if, for , implies either or .
Definition 15. A proper element is said to be 2-potent primary if, for , implies either or .
Theorem 16. Let a proper element be 2-potent primary. If is -primary for some , then is primary.
Proof. Clearly by Theorem 9, is -primary (almost primary). Let for some . If , then as is -primary we have or . If , then as is 2-potent primary we have or . Hence is primary.
Theorem 17. Let a proper element be -primary. If , then is primary.
Proof. Let for some . If , then as is -primary we have or . So assume . First suppose . Then for some in . Also and . As is -primary, either or for some . Hence or for some . Similarly if we can show that either or for some . So we can assume that and . Since , there exist in such that . Then , but . As is -primary, we have either or for some . Hence or for some . Thus is primary.
From the above theorem it follows that ① if a proper element is -primary but not primary, then ; ② a -primary element in with is primary.
Corollary 18. If a proper element is -primary but not primary, then .
Proof. From Theorem 17 we have . So which gives . Since , we have . Hence .
Corollary 19. If a proper element is -primary where , then is -primary.
Proof. If is primary, then by Theorem 10, is -primary. So assume that is not primary. Then by Theorem 17 and hypothesis we get . Hence for every . Consequently is -almost primary for every and hence it is -primary by Corollary 11.
Corollary 20. If a proper element is -primary but not primary, then .
Proof. The proof is obvious.
Now we obtain the characterization of a -primary element in .
Theorem 21. Let be a proper element and let be a function. Then the following statements are equivalent: ① is -primary; ② for every such that , either or ; ③ for any two elements , and implies either or .
Proof. ①②. Suppose ① holds. Let be compact such that and . Then . If , then . If , then since is -primary and it follows that . Hence by Lemma 1 of [6] either or . Consequently either or .
②③. Suppose ② holds. Let , , and for some . Then by ② we have either or . If , then as it follows that which contradicts . If , then gives .
③①. Suppose ③ holds. Let , , and for some . Then as is compactly generated, there exist such that , , and but , . Let be any compact element of . Then such that , , and . So by ③ which implies . Therefore, is -primary.
Like -primary element in we introduce the notion of -prime element in .
Definition 22. Let be a function. A proper element is said to be -prime (or briefly -prime) if, for , and implies either or .
Similarly -prime elements in are defined by following settings in the definition for -prime element. For any proper element ,(i) and then is called weakly prime element;(ii) and then is called almost prime element;(iii) and then is called -almost prime element.
The analogous theorems and corollaries (obtained for -primary elements in ) for -prime elements in are as follows whose proofs being on similar arguments are omitted.
Theorem 23. Let be functions such that . Then every proper -prime element in is -prime.
Theorem 24. Let be a proper element. Then is prime implies is weakly prime implies is -prime implies is -almost prime implies is almost prime.
Now we have the characterization of an -prime element in .
Corollary 25. Let be a proper element. Then is -prime if and only if is -almost prime for every .
Clearly every prime element in is -prime. But the converse is not true as shown in the following example by taking .
Example 26. Consider as in Example 6. Here the element is weakly prime and hence almost prime, while (0) is not prime since , but neither nor .
In the following successive three theorems we show conditions under which a -prime element in is prime.
The next theorem gives a characterization of a -prime element in .
Theorem 27. Let be a Noether lattice. Let be a non-nilpotent proper element satisfying the restricted cancellation law. Then is -prime for some if and only if is prime.
Theorem 28. Let a proper element be 2-potent prime. If is -prime for some , then is prime.
Theorem 29. Let a proper element be -prime. If , then is prime.
From the above theorem it follows that ① if a proper element is -prime but not prime, then ; ② a -prime element in with is prime.
Corollary 30. If a proper element is -prime but not prime, then .
Corollary 31. If a proper element is -prime where , then is -prime.
Corollary 32. If a proper element is -prime but not prime, then .
Theorem 33. Let be a proper element and let be a function. Then the following statements are equivalent: ① is -prime; ② for every such that , either or ; ③ for any two elements , and implies either or .
3. Almost Prime and Almost Primary Elements in
The study of almost prime and almost primary ideals for commutative rings is carried out by Bhatwadekar and Sharma [7] and Bataineh and Kuhail [4], respectively. From the previous section we know that almost prime and almost primary elements in are particular cases of -prime and -primary elements in , respectively. In this section, we obtain some results on an almost prime element and on an almost primary element in with characterizations.
The reader can verify the following statements. (1) Every almost prime element in is almost primary.
(Its converse holds for semiprime elements.) (2) Every prime element in is almost prime. (3) Every idempotent element in is almost prime. (4) Every prime element in is almost primary. (5) Every idempotent element in is almost primary.
But the converse is not true as shown in the following example.
Example 34. Consider the lattice of ideals of the ring . Then the only ideals of are the principal ideals . Clearly is a compactly generated multiplicative lattice. Here the element is almost primary but not idempotent. (6) Every primary element in is almost primary.
But the converse is not true, which is clear from Example 12.
In the following successive three theorems we show conditions under which an almost primary element in is primary.
Theorem 35. Let a proper element be 2-potent primary. Then being almost primary implies that is primary.
Proof. The proof is obvious.
Theorem 36. Let a proper element be 2-potent prime. Then ① being almost prime implies that is prime; ② being almost primary implies that is primary.
Proof. The proof is obvious.
From the following examples it is clear that ① an almost primary element in need not be 2-potent prime; ② a2-potent prime element in which is almost primary need not be prime.
Example 37. Consider as in Example 12. Here the element is almost primary but not 2-potent prime.
Example 38. Consider as in Example 34. Here the element is 2-potent prime, almost primary but not prime.
Theorem 39. Let be a local Noetherian domain. If a proper element is -almost primary , then is primary.
Proof. Let for some . If for all , then as is -almost primary we have or . If for all , then as is a local Noetherian domain, from Corollary 3.3 of [8], it follows that . Thus . Since is domain, we have either or which implies or and hence is primary.
Theorem 40. Let be a quasi-local Noether lattice. If a proper element is such that , then is almost primary.
Proof. Let and for some . If , then . So gives . Similarly gives . Now if , then and hence . Similarly gives . Hence in any case is almost primary.
The next theorem gives the characterization of an almost primary element in .
Theorem 41. Let a proper element be join irreducible. Then the following statements are equivalent: ① is almost primary; ② for every such that , ; ③ for every such that , either or .
Proof. ①②. Suppose ① holds. Let be compact such that and . Then . If , then . If , then since is almost primary and it follows that . Hence by Lemma 1 of [6] either or . But as , we have .
②③. Suppose ② holds. Let for some . Then by ② . Now as is join irreducible, we have either or .
③①. Suppose ③ holds. Let , , and for some . Then by ③ we have either or . If , then as it follows that which contradicts . If , then gives . Therefore is almost primary.
In view of Proposition 2 of [9] and the above Theorems 21 and 41 we have the following result.
Corollary 42. Let be a quasi-local Noether lattice. Let a proper element be weak meet principal and let be a nonzero weak join principal element in . Then the following statements are equivalent: ① is almost primary; ② for every such that , ; ③ for every such that , either or ; ④ for any two elements , and implies either or .
Proof. The proof is obvious.
The following theorem gives the similar characterization of an almost primary element in .
Theorem 43. A proper element is almost primary if and only if for every such that either or .
Proof. Assume that the proper element is almost primary. Let be compact such that and . Then . If , then . If , then since is almost primary and it follows that . Hence by Lemma 1 of [6] either or . But as , we have or . Conversely assume that for every such that either or . Let , , and for some . Then either or . If , then as it follows that which contradicts . If , then gives . Therefore is almost primary.
The analogous theorems and corollaries (obtained for almost primary elements in ) for almost prime elements in are as follows whose proofs being on similar arguments are omitted.
Theorem 44. Let be a local Noetherian domain. If a proper element is -almost prime , then is prime.
Theorem 45. Let be a quasi-local Noether lattice. If a proper element is such that , then is almost prime.
The next theorem gives the characterization of an almost prime element in .
Theorem 46. Let a proper element be join irreducible. Then the following statements are equivalent: ① is almost prime; ② for every such that , ; ③ for every such that , either or .
Corollary 47. Let be a quasi-local Noether lattice. Let a proper element be weak meet principal and let be a nonzero weak join principal element in . Then the following statements are equivalent: ① is almost prime; ② for every such that , ; ③ for every such that , either or ; ④ for any two elements , and implies either or .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors wish to thank the referee for useful comments.