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 S pec(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 S pec(G)(s)(M) with the Zariski topology from the viewpoint of being a Noetherian space.