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

K. Al-Zoubi, M. Jaradat

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. A proper graded submodule $N$ 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 graded ring $R$ to graded module $M$, called Zariski topology of $M$, and investigate several properties of the topology.