Let M be a lattice module over a C-lattice L. A proper element P of M is said to be classical prime if for a, b ∈ L and X ∈ M, abX ≤ P implies that aX ≤ P or bX ≤ P. The set of all classical prime elements of M , Spec cp (M) is called as classical prime spectrum. In this article, we introduce and study a topology on Spec cp (M), called as Zariski-like topology of M. We investigate this topological space from the point of view of spectral spaces. We show that if M has ascending chain condition on classical prime radical elements, then Spec cp (M) with the Zariski-like topology is a spectral space.

Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text]. In this paper, we introduce the concept of the Zariski second radical of elements of [Formula: see text] and investigate some properties of Zariski second radical of elements of [Formula: see text]. We also investigate when the Zariski second radical is equal to the second radical of elements of [Formula: see text]. We give a characterization of Noetherian space by using Zariski second radical of elements of [Formula: see text].