Classical Zariski Topology on Prime Spectrum of Lattice Modules

Let M be a lattice module over a  C-lattice L.  Let Spec^{p}(M) be the collection of all prime elements of M. In this article, we consider a  topology on Spec^{p}(M), called the classical Zariski topology and investigate the topological properties of Spec^{p}(M) and the algebraic properties of M. We investigate this topological space from the point of view of spectral spaces.  By  Hochster's characterization of a spectral space, we show that for each lattice module M with finite spectrum, Spec^{p}(M) is a spectral space. Also we introduce finer patch topology on Spec^{p}(M) and we show that Spec^{p}(M) with finer patch topology is a compact space and every irreducible closed subset of Spec^{p}(M) (with classical Zariski topology) has a generic point  and Spec^{p}(M) is a spectral space, for a lattice module M which has ascending chain condition on prime radical elements.

Keywords:

prime element , prime spectrum , classical Zariski topology , finer patch topology