The Graded Classical Prime Spectrum with the Zariski Topology as a Notherian Topological Space

Let 𝐺 be a group. Let R be a 𝐺-graded commutative ring with identity and let 𝑀 be a graded R-module. The graded classical prime spectrum 𝐶𝑙:Spec𝑔(𝑀) is de ned to be the set of all graded classical prime submodule of 𝑀. In this paper we estab- lish necessary and su cient conditions for 𝐶𝑙:Spec𝑔(𝑀) with the Zariski topology to be a Noetherian topological space.