Graded modules with Noetherian graded second spectrum

Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module. The set of all graded second submodules of $M$ is denoted by $Spe{c}_G^s(M),$ and it is called the graded second spectrum of $M$. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate $Spe{c}_G^s(M)$ with the Zariski topology from the viewpoint of being a Noetherian space.