In this paper, we study the concepts of n-absorbing, weakly nabsorbing and n-almost n-absorbing elements as a generalization of prime, weakly prime and almost prime elements in C-lattices.

Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

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LetLbe aC-lattice and letMbe a lattice module overL. Letϕ:M→Mbe a function. A proper elementP∈Mis said to beϕ-absorbing primary if, forx1,x2,…,xn∈LandN∈M,x1x2⋯xnN≤Pandx1x2⋯xnN≰ϕ(P)together implyx1x2⋯xn≤(P:1M)orx1x2⋯xi-1xi+1⋯xnN≤PM, for somei∈{1,2,…,n}. We study some basic properties ofϕ-absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules asϕ-absorbing elements andϕ-absorbing primary elements are unified.