We investigateϕ-prime andϕ-primary elements in a compactly generated multiplicative latticeL. By a counterexample, it is shown that aϕ-primary element inLneed not be primary. Some characterizations ofϕ-primary andϕ-prime elements inLare obtained. Finally, some results for almost prime and almost primary elements inLwith characterizations are obtained.

Abstract. In this paper we introduce and investigate 2-absorbing, n-absorbing, (n, k)-absorbing, weakly 2-absorbing, weakly n-absorbing and weakly (n, k)-absorbing elements in a lattice module M . Some characterizations of 2-absorbing and weakly 2-absorbing elements of M are obtained. By counter example it is shown that a weakly 2-absorbing element of M need not be 2-absorbing. Finally we show that if N ∈ M is a 2-absorbing element, then rad(N ) is a 2-absorbing element of M .Mathematics Subject Classification (2010): 06D10, 06E10, 06E99, 06F99

In this paper, we introduce φ-prime and φ-primary  elements in an L-module M. Many of its characterizations and properties are obtained. By counter examples, it is shown that a φ-prime element of M need not be prime, a φ-primary element of M need not be φ-prime, a φ-primary element of M need not be prime and a φ-primary element of M need not be primary. Finally, some results for almost prime and almost primary elements of an L-module M with their characterizations are obtained. Also, we introduce the notions of n-potent prime (respectively n-potent primary) elements in L and M to obtain interrelations among them where n≥2.

In this paper, we introduce the concepts of "An element prime to and primary to another element in lattice modules".Using these concepts,we generalize some results proved by C.S.Manjarekar and N.S.Chavan[4]. Some characterizations of an element prime to and primary to another element in lattice modules are obtained.It is shown that if a family of elements is primary to an element A then their supremum is also primary to A and if ∧ α B α is primary to an element A then each B α is primary to A.Finally,the characterization of an element non-primary to another element is obtained.