An Element Primary to Another Element

Abstract

"In this paper we have introduced the new concept ‘An element primary to another clement' and using this concept we have generalized some results proved by Anderson et al. Some results of join principally generated multiplicative lattices in which every semi-primary element is primary are proved. If S is a weak M-lattice, then the condition for a prime element to be maximal is proved. We have proved the following main results: If every prime element of L is a minimal prime element, then every completely meet semiprime element is primary. Suppose L is join principally generated and satisfying the condition (*). If a strong join principal element d is primary to b and p is a minimal prime over d ν b, then p is maximal in L. Suppose L is satisfying the condition (*). If a strong join principal element d is primary to b and p is a minimal prime over d ν b, then ∃ a p-primary element q such that d ≴ q and hence d ν b≴q.