%0 Journal Article
%T An Element Weakly Primary to Another Element
%A C. S. Manjarekar
%A U. N. Kandale
%J Chinese Journal of Mathematics
%D 2013
%R 10.1155/2013/495205
%X 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 £¿allialp 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). 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
%U http://www.hindawi.com/journals/cjm/2013/495205/