%0 Journal Article
%T Classical Zariski Topology on Prime Spectrum of Lattice Modules
%J Journal of Algebra and Related Topics
%I University of Guilan
%Z 2345-3931
%A Borkar, V.
%A Girase, P.
%A Phadatare, N.
%D 2018
%\ 12/01/2018
%V 6
%N 2
%P 1-14
%! Classical Zariski Topology on Prime Spectrum of Lattice Modules
%K prime element
%K prime spectrum
%K classical Zariski topology
%K finer patch topology
%R 10.22124/jart.2018.11106.1112
%X 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.
%U https://jart.guilan.ac.ir/article_3326_5f999e1eaeb83e79b53a441a2df5103f.pdf