Zariski topology on lattice modules

Let M be a lattice module over a C-lattice L and s(M) be the set of all prime elements in lattice modules M. 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 s(M) and algebraic properties of M.