Let R be a G-graded ring and M be a G-graded R-module. The graded second spectrum of M, denoted by SpecssG(M), is the set of all graded second submodules of M. In this paper, we define a topology on SpecssG(M) which is analogous to that for SpecG(R), and investigate several topological properties of this topology.

<abstract><p>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 $ Spec_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 $ Spec^s_G(M) $ with the Zariski topology from the viewpoint of being a Noetherian space.</p></abstract>

Let R be a G-graded ring and M be a G-graded R-module. We define the graded primary spectrum of M, denoted by PSG(M), to be the set of all graded primary submodules Q of M such that (GrM(Q) :RM) = Gr((Q:RM)). In this paper, we define a topology on PSG(M) having the Zariski topology on the graded prime spectrum SpecG(M) as a subspace topology, and investigate several topological properties of this topological space.

Let R be a G-graded ring and M be a G-graded R-module. We define the graded primary spectrum of M , denoted by PSG(M ), to be the set of all graded primary submodules Q of M such that (GrM (Q) :R M ) = Gr((Q :R M )). In this paper, we define a topology on PSG(M ) having the Zariski topology on the graded prime spectrum SpecG(M ) as a subspace topology, and investigate several topological properties of this topological space.