CLASSICAL ZARISKI TOPOLOGY ON PRIME SPECTRUM OF LATTICE MODULES

Let M be a lattice module over a C-lattice L. Let Specp (M) be the collection of all prime elements of M. In this article, we consider a topology on Specp (M), called the classical Zariski topology and investigate the topological properties of Specp (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, Specp (M) is a spectral space. Also we introduce finer patch topology on Specp (M) and we show that Specp (M) with finer patch topology is a compact space and every irreducible closed subset of Specp (M) (with classical Zariski topology) has a generic point and Specp (M) is a spectral space, for a lattice module M which has ascending chain condition on prime radical elements.