Some topological properties of the Zariski topology on \(PL.Spec_{g}(\Im )\)

Let G be a group with identity e. Let \(\Re \) $\Re$ be a G-graded commutative ring and \(\Im \) $\Im$ a graded \(\Re \) $\Re$ -module. The graded primary-like spectrum \( PL.Spec_{g}(\Im )\) $PL.Spe{c}_g(\Im )$ is defined to be the set of all graded primary-like submodules of \(\Im \) $\Im$ satisfying the gr-primeful property. The Zariski topology on \(PL.Spec_{g}(\Im )\) $PL.Spe{c}_g(\Im )$ , denoted by \(PL.\tau ^{g}\) $PL.{\tau }^g$ , is described by taking the set PL- \(\eta (\Im )=\{PL\) $\eta (\Im )=\{PL$ - \(V_{\Im }^{g}({\mathcal {C}})\mid {\mathcal {C}}\) $V_{\Im }^g(\mathcal{C})\mid \mathcal{C}$ is a graded submodule of \(\Im \}\) $\Im \}$ as the set of closed sets of \(PL.Spec_{g}(\Im ),\) $PL.Spe{c}_g(\Im ),$ where PL- \(V_{\Im }^{g}({\mathcal {C}})=\{P\in PL.Spec_{g}(\Im )\mid Gr((P:_{\Re }\Im ))\supseteq Gr(({\mathcal {C}}:_{\Re }\Im ))\}.\) $V_{\Im }^g(\mathcal{C})=\{P\in PL.Spe{c}_g(\Im )\mid Gr((P{:}_{\Re }\Im ))\supseteq Gr((\mathcal{C}{:}_{\Re }\Im ))\}.$ In this paper, we study the irreducible and irreducible components closed subsets of ( \(PL.Spec_{g}(\Im )\) $PL.Spe{c}_g(\Im )$ , \(PL.\tau ^{g}\) $PL.{\tau }^g$ ) and we give a result about Noetherianness of the graded primary-like spectrum of a graded module. Also, we study the topological space ( \(PL.Spec_{g}(\Im )\) $PL.Spe{c}_g(\Im )$ , \( PL.\tau ^{g}\) $PL.{\tau }^g$ ) from the point of view of spectral space.