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Chinese Journal of Mathematics
Volume 2013 (2013), Article ID 495205, 4 pages
http://dx.doi.org/10.1155/2013/495205
Research Article

An Element Weakly Primary to Another Element

1Department of Mathematics, Shivaji University, Kolhapur, India
2Department of Mathematics, Sharad Institute of Technology, College of Engineering, Yadrav, Ichalkaranji, India

Received 3 August 2013; Accepted 25 August 2013

Academic Editors: Z.-Y. Li, Y. Ouyang, and W. Zhu

Copyright © 2013 C. S. Manjarekar and U. N. Kandale. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the concept “An element weakly primary to another element” and using this concept we have generalized some result proved by Manjarekar and Chavan (2004). It is shown that if is a family of elements weakly primary to a in L, then is weakly primary to a.

1. Introduction

Multiplicative lattice is a complete lattice provided with commutative, associative, and join distributive multiplication for which the largest element 1 acts as a multiplicative identity. A proper element of is called prime element if  or  for and is called primary element if implies or for some . An element of is called compact if , and implies the existence of finite number of elements of such that . Throughout this paper, denotes compactly generated multiplicative lattice with 1 compact and every finite product of compact elements is compact. Let be the set of all compact elements in . Also, is the greatest element in such that . An element is join principal if and meet principle if , for all  .

An element is principle if it is both join and meet principle. For , . An element is called semiprimary if is primary element. is said to satisfy the condition (*) if every semiprimary element is primary element.

An element is said to be strong join principle element if is compact and join principle. An element is p-primary if is primary and and is semiprime if . An element of is called zero divisor if such that , and if has no zero divisor then will be called lattice domain or simply a domain. denotes the set of compact elements of .

The concept of weakly prime element is studied by Çallialp et al. [1]. The concept of weakly primary element is introduced by Sachin and Vilas [2]. For other definitions and simple properties of multiplicative lattice, one can refer to Dilworth [3].

Definition 1. Weakly primary element is defined as follows.

An element is said to be a weakly primary element if for , implies or for some .

Example 2. Lattice of ideals of ring (see Figure 1).

495205.fig.001
Figure 1

In the lattice of Example 2, an element is weakly primary element. From Definition 1, it is clear that every weakly prime element is weakly primary element Converse need not be true. Since in Example 2,   is weakly primary element but it is not weakly prime element. Further, if is a weakly primary element, then is a weakly prime element. Because if for compact element and such that then for some . As is a weakly primary element, either or for some . Consequently, or . Thus, is a weakly prime element. This implies that every weakly primary element is a weakly semiprimary element. It need not be true that is always weakly prime or is always weakly semiprimary. In Example 2, the least element 0 is not semiprimary as is not a weakly prime element. The concept of “An element prime to another element” is introduced in [4]. An element is prime to an element if for , implies .

Now, we define the following.

Definition 3. An element weakly prime to another element is defined as follows.
An element, , is called weakly prime to an element if for any , implies .
In Example 2, the element is weakly prime to an element , but is not weakly prime to any other element of . This follows directly from the fact that an element is weakly prime to an element if and only if .

2. An Element Weakly Primary to Another Element

Now we introduce the following main concept which is a generalization of the concept introduced by Manjarekar and Chavan [5].

Definition 4. An element is said to be weakly primary to another element in if for , implies for some .

In Example 2, the element is weakly primary to , but note that is not weakly prime to . This follows directly from Corollary 9, and note that and . Evidently, if is weakly prime to , then is weakly primary to in . Now if is a weakly prime element and , then is weakly prime to , and if is a weakly primary element and a compact element , then is a weakly primary element to .

Thus, from this, it is clear that elements weakly primary to another element exist in the lattice . Since is compactly generated multiplicative lattice with 1 compact, weakly prime element and hence weakly primary element exists in . Hereafter, will be a domain. We prove some interesting results including characterizations.

Theorem 5. No proper nonzero element is weakly prime or weakly primary to itself in .

Proof. If is a proper nonzero element in and is weakly primary to itself, then implies that , a contradiction. Therefore, no proper nonzero element is weakly prime or weakly primary to itself in .

Now we prove some characterizations of an element weakly primary to .

Theorem 6. Let be a semiprime element. Then is weakly primary to if and only if is weakly prime to .

Proof. Assume that is weakly primary to semiprime element . Let for some . Then for some . Consequently, . As is semiprime, . Thus, is weakly prime to . The converse part is obvious.

Theorem 7. Let be a lattice domain. Let ; then is weakly primary to if and only if .

Proof. Assume that is weakly primary to . Let such that ; then, . As is weakly primary to for some . Hence, . This shows that .
Conversely, assume that . Let for some . Then, we have . This implies that for some . Thus, is weakly primary to .

Theorem 8. Let be a lattice domain. Let ; then is weakly prime to if and only if .

Proof. Assume that is weakly prime to in . Let such that . Then . As is weakly prime to , we have . This shows that . But . Therefore, we get .
Conversely, assume that . Let, for . Then, we get . Thus is weakly prime to .

Corollary 9. Let be a lattice domain. Let, . Then, is weakly primary to but it is nonweakly prime to if and only if .

Proof. It follows from the fact that and from Theorems 7 and 8.

Corollary 10. Let . If a is weakly semiprimary element and , then is weakly primary to a.

Proof. Assume that is a weakly semiprimary element and . Let such that . Then . As is a weakly prime element and , we have . This implies that for some . Thus, is weakly primary to .

Theorem 11. Let be a lattice domain. Let . Then, is weakly prime to if and only if b is weakly prime to for every .

Proof. Assume that is weakly prime to in . Let for some . Then, . As is weakly prime to , . Consequently, . Thus, is weakly prime to for every . The converse is obvious, since, if is weakly prime to for every , is weakly prime to .

Theorem 12. Let and let be a lattice domain. If is weakly primary to and is a semiprime element in , then is weakly primary to for every .

Proof. It follows from Theorems 6 and 11.

Theorem 13. Let . Then is weakly primary to in if and only if each is weakly primary to .

Proof. Assume that is weakly primary to in . Let and for some . Then . Therefore, by assumption, for some .
This shows that each is weakly prime to . The converse part is obvious.

Theorem 14. If is a family of a elements weakly primary to in , then is weakly primary to .

Proof. It follows from the fact that and from Theorem 13.

Theorem 15. Let . Then, is nonweakly primary to if and only if each is nonweakly primary to .

Proof. Assume that is nonweakly primary to . Therefore, by Theorem 8, we have . Let be an element of such that . Then . This shows that . Thus, again by Theorem 8, each is nonweakly primary to .

This lemma leads us to the following two obvious corollaries.

Corollary 16. If is a family of elements nonweakly primary to in , then is nonweakly primary to .

Corollary 17. If is nonweakly primary to in , then each is nonweakly primary to .

Theorem 18. If y is compact and is nonweakly primary to a, then either is nonweakly primary to or is nonweakly primary to for some .

Definition 19. Completely meet semiprimary elements are defined as follows.
An element is said to be completely meet semiprimary element if is a completely meet prime element.

Example 20. Every element is not a completely meet prime element. But note that . Thus, each is a completely meet semiprimary element (see Figure 2).

495205.fig.002
Figure 2

Result 1. In any multiplicative lattice , we have for any .

Theorem 21. Suppose that is a completely meet semiprimary element of and let . If each is nonweakly primary to in , then is nonweakly primary to a.

Proof. By Result 1, we have . But each is nonweakly primary to . Consequently, by Theorem 12, each . As is completely meet semiprimary element, . Therefore, by Theorem 8, is nonweakly primary to .

Now we construct a new element as follows. Define .

Theorem 22. If is a completely meet semiprimary element, then is weakly primary element and is nonweakly primary to .

Proof. By Theorem 21, it follows that is nonweakly primary element to . Let and be compact elements such that, . Then by Theorem 15, is nonweakly primary to . Therefore, by Theorem 18, either is nonweakly primary to or is nonweakly primary to for some . Thus, or for some . This shows that is a weakly primary element.

References

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