ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES

Authors : Pradip GİRASE, Vandeo BORKAR, Narayan PHADATARE

Pages : 186-198

Doi:10.24330/ieja.504147

View : 20 | Download : 8

Publication Date : 2019-01-08

Article Type : Research Paper

Abstract :Let $M$ be a lattice module over a $C$-lattice $L$. A proper element $P$ of $M$ is said to be classical prime if for $a ,b\in L$ and $X\in M, abX\leq P$ implies that $aX\leq P$ or $bX\leq P$. The set of all classical prime elements of $M$, $Spec^{cp}insert ignore into journalissuearticles values(M);$ is called as classical prime spectrum. In this article, we introduce and study a topology on $Spec^{cp}insert ignore into journalissuearticles values(M);$, called as Zariski-like topology of $M$. We investigate this topological space from the point of view of spectral spaces. We show that if $M$ has ascending chain condition on classical prime radical elements, then $Spec^{cp}insert ignore into journalissuearticles values(M);$ with the Zariski-like topology is a spectral space.  
Keywords : Classical prime element, classical prime spectrum, classical prime radical element, Zariski like topology, spectral space