Residuation Properties and Weakly Primary Elements in Lattice Modules

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