TY - JOUR ID - 3326 TI - Classical Zariski Topology on Prime Spectrum of Lattice Modules JO - Journal of Algebra and Related Topics JA - JART LA - en SN - 2345-3931 AU - Borkar, V. AU - Girase, P. AU - Phadatare, N. AD - Department of Mathematics, Yeshwant Mahavidyalaya, Nanded, India AD - Department of Mathematics, K K M College, Manwath, Dist- Parbhani. 431505. Maharashtra, India. AD - Department of Mathematics, Savitribai Phule Pune University, Pune. Maharashtra. India Y1 - 2018 PY - 2018 VL - 6 IS - 2 SP - 1 EP - 14 KW - prime element KW - prime spectrum KW - classical Zariski topology KW - finer patch topology DO - 10.22124/jart.2018.11106.1112 N2 - 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. UR - https://jart.guilan.ac.ir/article_3326.html L1 - https://jart.guilan.ac.ir/article_3326_5f999e1eaeb83e79b53a441a2df5103f.pdf ER -