The Zariski topology on the graded classical prime spectrum of a graded module over a graded commutative ring

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a $G$-gradedcommutative ring and $M$ a graded $R$-module. A proper gradedsubmodule $N$ of $M$ is called a graded classical prime if whenever$r,s\in h(R)$ and $m\in h(M)$ with $rsm\in N$, then either $rm\in N$or $sm\in N$. The graded classical prime spectrum $Cl.Spec^{g}(M)$is defined to be the set of all graded classical prime submodules of$M$. In this paper, we introduce and study a topology on$Cl.Spec^{g}(M)$, which generalizes the Zariski topology of gradedring $R$ to graded module $M$, called Zariski topology of $M$, andinvestigate several properties of the topology.