Residuation Properties and Weakly Primary Elements in Lattice Modules

Abstract

We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice module and weakly prime element of a multiplicative lattice .

1. Introduction

A multiplicative lattice is a complete lattice provided with commutative, associative, and join distributive multiplication in which the largest element acts as a multiplicative identity. An element is called proper if . A proper element of is said to be prime if implies or . If , , is the join of all elements in such that . A proper element of is said to be primary if implies or for some positive integer . If , the radical of a is denoted by . An element is called compact if implies for some finite subset . Throughout this paper, denotes a compactly generated multiplicative lattice with 1 compact and every finite product of compact elements is compact. We will denote by the set compact elements of . A nonempty subset of is called a filter of if the following conditions are satisfied:(1) implies ;(2), implies .

Let denote the set of all filters of . For a nonempty subset , define , for some . Then it is observed that is a complete distributive lattice with as the supremum and the set theoretic as the infimum. For the smallest filter containing a is denoted by and it is given by for some nonnegative integer . For a filter we denote .

Let be a complete lattice and a multiplicative lattice. Then is called -module or module over if there is a multiplication between elements of and written as where and which satisfies the following properties:(1),  , ;(2),  , ;(3), , ;(4);(5), for all and , where is the supremum of and is the infimum of . We denote by and the least element and the greatest element of . Elements of will generally be denoted by and elements of will generally be denoted by .

Let be an -module. If and then . If , then . An -module is called a multiplication -module if for every element there exists an element such that . In this paper a lattice module will be a multiplication lattice module. A proper element of is said to be prime if implies or ; that is, for every , . If is a prime element of then is prime element of [1]. An element in is said to be primary if implies or ; that is, for some integer . An element of is called a radical element if .

2. Residuation Properties

We prove some elementary properties of residuation. Such results are obtained by Anderson and others [2] for compactly generated multiplicative lattices.

Theorem 1. Let L be a multiplicative lattice and a multiplication lattice module over . For and , where is a radical element, one has the following identities:(i) implies and ;(ii);(iii);(iv) for every ;(v);(vi) implies for every ;(vii) implies ;(viii)For in , ;(ix) implies ;(x);(xi) for some positive integer .

Proof. (i)  We have . As , , so implies . This shows that . Next part is obvious.
(ii)  We have . This implies that .
(iii)  We have, by (ii), . So by (i), . If , . So, . Hence, implies . Therefore, .
(iv)  We have . As , we have for every . Let where . As is a multiplication lattice module for some . So . Hence . So ; that is, , and hence for each positive integer .
(v)  Note that . Therefore by (i), . Let . Now we have and . Then, and . This implies that , . Therefore, we have [3]. Hence, , by (iv). That is, or . Consequently, . Note that . Then by (i), we have . Let . We have , by (iv). This shows that . Thus, . Consquently, .
(vi)  Assume that . We have, by (iv), for every . But . Thus, for every .
(vii)  Assume that . As , we have , . Therefore, . Let in be such that . Then, . This gives and . Then we have, for in , . Consequently, For the next part, where . We have , . Therefore, we get . Let be an element of such that . Then, . This gives and . Then we have . Consequently, .
(viii)  We have, for in ,   and . This implies that . This shows that, for in , .
(ix)  Let . We have implies . implies . So, . Therefore, .
(x)  As , by (ii), we get . But . Therefore, . Thus, .
(xi)  As , we have . So . Let where . Then ; that is, . Therefore, . Hence, , since is a radical element. This implies that . Consequently, . Hence, .

Let denote the set of compact elements of . We define for some . Such concept is defined for multiplicative lattices by Anderson and others [2].

We obtain the necessary and sufficient condition for an element of a lattice module to be contained in .

Theorem 2. Let be a filter of and let be a compact element of . Then if and only if for some .

Proof. As is compact element and , there exist finite number of elements such that where for some . Let . Then by hypothesis, finite product of compact elements is compact, and hence, is a compact element. As is a filter, . Also, for each . Hence, we have . Conversely, suppose for some . Hence, obviously.

Now we obtain an equivalent formulation for .

Theorem 3. For , .

Proof. Let . Let be a compact element of such that . Then by Theorem 2 there exists such that . So, . Hence, . Therefore, . Let be a compact element such that . Then, there exist such that . By compactness of , there exist compact elements such that and . As is a filter, we have . Hence, . Thus, . This shows that .

Theorem 4. For (1) implies ;(2).

Proof. (1)  Let and let be a compact element of such that . Then by Theorem 2  , for some . By assumption, , . Hence, and .
(2)  We have . Hence by 1, . Thus, . Let be a compact element in such that . Then and . By Theorem 2, there exist elements and such that and . So and . Consequently, [3]. As and are compact elements, is also compact. As and are filters, and . This implies that . Thus, where . That is, . Hence, by Theorem 2, . That is, . Thus, .

3. Weakly Prime and Weakly Primary Elements

We introduce the concepts of weakly prime and weakly primary elements in lattice modules. These concepts are generalisations of weakly prime elements in multiplicative lattices [4] and weakly primary ideals [5].

Definition 5. A proper element of is said to be weakly prime element if implies or for every , .

Definition 6. A proper element of is said to be weakly primary element if implies or for every , .

Example 7. Let denote a cyclic -module and let denote the set of all ideals of and let denote the set of all submodules of . Then is a lattice module over . Let . Then is a weakly prime element of .

Example 8. Let denote the set of integers and let denote the set of all ideals of . Then is a multiplicative lattice. Let . Then is a module over . Let denote the set of all submodules of . Then is a lattice module over . Let ; then is a weakly primary element of ; however it is not weakly prime element of , because . But neither nor ; see [6].

The next result gives the relation between weakly primary element of and weakly primary element of .

The study of such concepts for submodules is carried out by Eid [6].

Theorem 9. Let be a multiplicative lattice, a lattice module over , and a weakly primary element of . Then is a weakly primary element of .

Proof. Suppose is a weakly primary element of . Let and where . This implies that . But is a weakly primary element and and . So for some ; that is, . Hence, is a weakly primary element of .

Now we obtain the relation between weakly primary element of and weakly prime element of where is assumed to be without divisors of zero.

Theorem 10. Let be a multiplicative lattice and a lattice module over and suppose is a weakly primary element of . Then is a weakly prime element of .

Proof. Suppose is a weakly primary element of . Then by Theorem 9   is a weakly primary element of . Let with . Hence for any . That is, for any . We have for some . That is, where and is a weakly primary element . Hence, for some . That is, where . Thus is a weakly prime element of .

Conflict of Interests

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