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Classical Zariski Topology on Prime Spectrum of Lattice Modules | ||
| Journal of Algebra and Related Topics | ||
| مقاله 1، دوره 6، شماره 2، اسفند 2018، صفحه 1-14 اصل مقاله (320.74 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2018.11106.1112 | ||
| نویسندگان | ||
| V. Borkar1؛ P. Girase* 2؛ N. Phadatare3 | ||
| 1Department of Mathematics, Yeshwant Mahavidyalaya, Nanded, India | ||
| 2Department of Mathematics, K K M College, Manwath, Dist- Parbhani. 431505. Maharashtra, India. | ||
| 3Department of Mathematics, Savitribai Phule Pune University, Pune. Maharashtra. India | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| prime element؛ prime spectrum؛ classical Zariski topology؛ finer patch topology | ||
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