Second and Secondary Lattice Modules

Abstract

Let M be a lattice module over the multiplicative lattice L. A nonzero L-lattice module M is called second if for each , or . A nonzero L-lattice module M is called secondary if for each , or for some . Our objective is to investigative properties of second and secondary lattice modules.

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A multiplicative lattice is a complete lattice in which there is defined as a commutative, associative multiplication which distributes over arbitrary joins and has the compact greatest element (least element ) as a multiplicative identity (zero). An element is said to be proper if . An element in is said to be prime if implies either or . If is prime, then is said to be a domain. For , we define . An element in is said to be primary if implies either or .

If , belong to is the join of all such that . An element of is called meet principal if for all . An element of is called join principal if for all . is said to be principal if is both meet principal and join principal. is said to be weak meet (join) principal if for all . An element of a multiplicative lattice is called compact if implies for some subsets . If each element of is a join of principal elements of , then is called a -lattice (CG-lattice). If is a -lattice and is a primary element, then is prime [1, Lemma 2.1].

Let be a complete lattice. Recall that is a lattice module over the multiplicative lattice or simply an -module in case there is a multiplication between elements of and , denoted by for and , which satisfies the following properties:(1);(2);(3);(4);for all , , in and for all , in .

Let be an -module. If , belong to is the join of all such that . Particularly, is denoted by . If and , then is the join of all such that . An element of is called meet principal if for all and for all . An element of is called join principal if for all and for all . is said to be principal if it is both meet principal and join principal. In special cases, an element of is called weak meet principal if for all . is said to be weak principal if is both weak meet principal and weak join principal.

Let be an -module. An element in is called compact if implies for some subsets . The greatest element of will be denoted by . If each element of is a join of principal elements of , then is called a -lattice module (-lattice module).

Let be an -module. An element is said to be proper if . For all elements of , is a set of all such that and is an -lattice module with for all and such that .

For various characterizations of lattice modules, the reader is referred to [2–9].

Definition 1. A nonzero -lattice module is called second if for each , or .

Definition 2. A nonzero -lattice module is called secondary if for each , or for some .

Example 3. Let be the integers, let be the rational numbers, and let be -module. Suppose is the set of all ideals of and is the set of all submodules of . Thus, as -lattice module is a second module, since for every integer , or .

Remark 4. Every second lattice module is a secondary lattice. But the converse is not true. For this, we can give the following example.

Example 5. Let be the integers and let be -module. Suppose that is the set of all ideals of and is the set of all submodules of . Thus, as -lattice module is a secondary lattice module, which is not a second lattice module.

Example 6. Let be the integers and the set of all ideals of . Thus, as -lattice module is neither a second lattice module nor a secondary lattice module.

Proposition 7. Let be a -lattice and let be a nonzero -lattice module. If for each compact , or , then is a second -lattice module.

Proof. Let . Since is a -lattice, then we have such that ’s are compact elements of . Then, we obtain . We have two cases.
Case 1. If for each compact , then we have .
Case 2. If for some compact , then we have .
Hence, or for each . Consequently, is second.

Proposition 8. If is a second -lattice module, then is a prime element of . In this case, is called -second lattice module.

Proof. Suppose that is a second -lattice module. Clearly, is a proper element of . Let and assume that ; that is, . But is a second -lattice module; then . Since and , then , which implies that .

Proposition 9. If is a secondary -lattice module, then is a primary element of .

Proof. Suppose that is a secondary -lattice module. Let and ; we prove that . Since and , we have and for each . Since is secondary, we have . Then , which implies .

Proposition 10. Let be a -lattice. If is a secondary -lattice module, then is a prime element of . In this case, is called -secondary lattice module.

Proof. Let be a secondary lattice. Then is a primary element of by Proposition 9. Therefore is prime by [1, Lemma 2.1].

Proposition 11. Let be a lattice domain and let be a nonzero -module. Then is a second lattice module with if and only if is a secondary lattice module with .

Proof. Since is a second lattice module, then is a secondary lattice module. Since is domain, then .
Suppose that is a secondary lattice module with . Let and assume that . Since is a secondary lattice module, then there exists a positive integer such that ; that is, . Then . Hence, we obtain is a second lattice. Clearly, .

Definition 12. Let be a nonzero -lattice module. An element is said to be pure element, if for all .

Proposition 13. Let be a -lattice, let be a nonzero -lattice module, and let be a pure element of . If is a -secondary lattice module, then and are both -secondary lattice modules.

Proof. Suppose that is a -secondary lattice module. Let . Since is a secondary lattice module, then either and in this case or there exists a positive integer , such that and in this case . Hence, is a secondary lattice module.
It remains to show that . Clearly, . Let be compact and . Since is compact, there exists a positive integer such that ; that is, . Hence, we have . Now we assume that . Then , since is secondary. Thus, ; that is, , which is a contradiction. Therefore, . Consequently, we obtain .
Let . Since is pure, . As is a secondary lattice module, then either or there exists a positive integer such that . This implies that either or . Therefore, we have or . Hence, is a secondary lattice module.
Now we show that . Clearly, . Let be compact and . Since is compact, there exists a positive integer such that . Since is pure, we have . If , then . Thus, . This implies that , a contradiction. Therefore, .

Proposition 14. Let be a nonzero -lattice module and let be a pure element of . Then is a -second lattice module if and only if and are both -second lattice modules.

Proof. Suppose that is a -second lattice module. Let . Since is pure, we have . As is a second lattice module, then either or . This implies that either or . Hence, is a second lattice module. Now, we show that . Clearly, . Let . Thus, we have . Now we assume that . Then we obtain , since is a second lattice module. This implies that , a contradiction. Therefore, .
Let . Since is a second lattice module, either and in this case or and in this case . Hence, is a second lattice module. It remains to show that . Clearly . Let . Thus, ; that is, , and this implies that . Now we suppose that . Then, we have , since is second. Hence, , a contradiction. Therefore, .
Suppose that and are both second lattice modules with . Let . We have two cases.
Case 1. If , then and , which implies . Thus, .
Case 2. If , then since is a second lattice module. Hence, we have , that is, , since is pure. Because is a second lattice module and ; we obtain . Therefore, we obtain that . Consequently, is a second lattice module.
Now we show that . Clearly . Let . Then we have , that is; . Thus, , and so . Now, we assume that . Then, we have , since is second. Hence, , a contradiction. Consequently, we have .

Definition 15. An -module is called a multiplication lattice module if for every element , there exists an element , such that .

Definition 16. A element of an -module is called prime element if and whenever and with , then or .

Definition 17. A element of an -module is called semiprime element if and whenever and with , then .

Remark 18. Let be a proper element of an -module . Then is a semiprime element if and only if whenever , and is a positive integer with , then .

We know that a prime element is semiprime, but the converse is not true in general. The following proposition shows that the converse is true when the module is secondary and multiplication.

Proposition 19. Let be a multiplication and secondary -lattice module. For all element of such that , is a semiprime element of if and only if is a prime element of .

Proof. Suppose that is a semiprime element of and let , where , . Since is a secondary lattice module, then either for some positive integer or .
Case  1. If , then . Since is a semiprime element, we have .
Case  2. If , then we have , since is a multiplication lattice module. Then we have .
Therefore, is a prime element of .
It is obvious.

Definition 20. Let be an -lattice module and let be a proper element of . is called a primary element of , if whenever , such that , then or . Particularly, if is nonzero and is primary, then is said to be primary lattice module.

Definition 21. An -lattice module is said to be simple lattice module if .

Proposition 22. Every multiplication secondary lattice module is a primary lattice module.

Proof. Let be a multiplication secondary module and for some . Now, we assume that . Since is a secondary module, then we have . Because is a multiplication, then we have . Consequently, we obtain .

Proposition 23. Every multiplication second lattice module is a simple lattice module.

Proof. Let be a multiplication and second module. Since is a multiplication, for every , there exists such that . Then we obtain or , since is second. Thus, we have or for every ; that is, is simple.

Definition 24. Let be a domain and let be a nonzero -lattice module. If for every , then is said to be divisible.

Definition 25. A nonzero -lattice module is said to be torsion if there exists such that .

Proposition 26. Let be a domain. Let be a secondary -lattice module. Then either is a divisible module or is a torsion module.

Proof. Suppose that is a secondary module over a domain . If is not divisible, then there exists such that . Since is a secondary lattice module, then there exists a positive integer such that . Since and is a domain, then we have . Consequently, there exists such that . Therefore, is a torsion lattice module.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.