Some topological properties of the Zariski topology on \(PL.Spec_g (\Im)\) (Q6566481)

Suppose that \(G\) is a group with identity e, \(\mathfrak{R}\) is a \(G\)-graded commutative ring, and \(\mathfrak{J}\) a graded \(\mathfrak{R}\)-module. Then the \textit{graded primary-like spectrum} \(\mathrm{PL.Spec}_g(\mathfrak{J})\) is the set of all graded primary-like submodules of \(\mathfrak{J}\) satisfying the \(gr\)-primeful property.\N\NRecall that the \textit{Zariski topology} on \(\mathrm{PL.Spec}_g(\mathfrak{J})\), which is denoted by \(\mathrm{PL.}\tau^g\), is defined by taking the set (PL-\(\eta(\mathfrak{J}) = \{\mathrm{PL-}V^{g}_{\mathfrak{J}}(\mathcal{C}) \mid \mathcal{C} \; \text{is a graded submodule of }\; \mathfrak{J}\}\) as the set of closed sets of \(\mathrm{PL.Spec}_g(\mathfrak{J})\), where PL-\(V^g_{\mathfrak{J}}(\mathcal{C})=\{P \in \mathrm{PL.Spec}_g(\mathfrak{J}) \mid Gr((P :_{\mathfrak{R}} \mathfrak{J})) \supseteq Gr((\mathcal{C} :_{\mathfrak{R}} \mathfrak{J}))\}.\) The main aim of this work is to explore the topological properties of the Zariski topology on \(\mathrm{PL.Spec}_g(\mathfrak{J})\). The authors, in Section 2, focus on the irreducible closed subsets of the Zariski topology over \(\mathrm{PL.Spec}_g(\mathfrak{J})\) and their generic points (see Theorem 2.9 and Theorem 2.14). Next, in Section 3, they discuss the basic properties of graded Zariski primary-like radical and graded Zariski-Top primary-like radical of a graded submodule.\N\N Finally, in Section 4, the authors try to present some results related to Noetherianness of the graded primary-like spectrum of a graded module (refer to Theorems 4.1, 4.2, 4.5, 4.6, and 4.7) In particular, the authors, in Theorem 4.13, explore the topological space \((\mathrm{PL.Spec}_g(\mathfrak{J}), \mathrm{PL.}\tau^g)\) from the point of view of spectral space.