University of Guilan Journal of Algebra and Related Topics 2345-3931 6 2 2018 12 01 Classical Zariski Topology on Prime Spectrum of Lattice Modules 1 14 3326 10.22124/jart.2018.11106.1112 EN V. Borkar Department of Mathematics, Yeshwant Mahavidyalaya, Nanded, India P. Girase Department of Mathematics, K K M College, Manwath, Dist- Parbhani. 431505. Maharashtra, India. 0000-0002-7847-8296 N. Phadatare Department of Mathematics, Savitribai Phule Pune University, Pune. Maharashtra. India Journal Article 2018 08 17 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. https://jart.guilan.ac.ir/article_3326_5f999e1eaeb83e79b53a441a2df5103f.pdf