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 -