Zariski-like topologies for lattices with applications to modules over associative rings

We study Zariski-like topologies on a proper class X L of a complete lattice = (L, 0, 1). We consider X with the so-called classical Zariski topology (X,τcl) and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that is X-top if τ:= {X\V (a)|a L},where V (a) = {x L|a ≤ x} is a topology. We study the interplay between the algebraic properties of an X-top complete lattice and the topological properties of (X,τcl) = (X,τ). Our results are applied to several spectra which are proper classes of := LAT(RM) where M is a nonzero left module over an arbitrary associative ring R (e.g. the spectra of prime, coprime, fully prime submodules) of M as well as to several spectra of the dual complete lattice 0 (e.g. the spectra of first, second and fully coprime submodules of M).