The Graded Classical Prime Spectrum with the Zariski Topology as a Notherian Topological Space
Abstract: Let G be a group. Let R be a G-graded commutative ring with identity and let M be a graded R-module. The graded classical prime spectrum Cl.Spec g (M ) is defined to be the set of all graded classical prime submodule of M . In this paper we establish necessary and sufficient conditions for Cl.Spec g (M ) with the Zariski topology to be a Noetherian topological space.
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The quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring
Let G be a group. Let R be a G-graded commutative ring and let M be a graded R-module. A proper graded submodule Q of M is called a graded quasi-primary submodule if whenever r ∈ h ( R ) {r\in h(R)} and m ∈ h ( M ) {m\in h(M)} with r m ∈ Q {rm\in Q} , then either r ∈ Gr ( ( Q : R M ) ) {r\in\operatorname{Gr}((Q:_{R}M))} or m ∈ Gr M ( Q ) {m\in\operatorname{Gr}_{M}(Q)} . The graded quasi-primary spectrum qp . Spec g ( M ) {\mathop{\rm qp.Spec}\nolimits_{g}(M)} is defined to be the set of all graded quasi-primary submodules of M. In this paper, we introduce and study a topology on qp . Spec g ( M ) {\mathop{\rm qp.Spec}\nolimits_{g}(M)} , called the quasi-Zariski topology, and investigate the properties of this topology and some conditions under which ( qp . Spec g ( M ) , q . τ g ) {(\mathop{\rm qp.Spec}\nolimits_{g}(M),q.\tau^{g})} is a Noetherian, spectral space.
The quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring
Let G be a group. Let R be a G-graded commutative ring and let M be a graded R-module. A proper graded submodule Q of M is called a graded quasi-primary submodule if whenever r ∈ h ( R ) {r\in h(R)} and m ∈ h ( M ) {m\in h(M)} with r m ∈ Q {rm\in Q} , then either r ∈ Gr ( ( Q : R M ) ) {r\in\operatorname{Gr}((Q:_{R}M))} or m ∈ Gr M ( Q ) {m\in\operatorname{Gr}_{M}(Q)} . The graded quasi-primary spectrum qp . Spec g ( M ) {\mathop{\rm qp.Spec}\nolimits_{g}(M)} is defined to be the set of all graded quasi-primary submodules of M. In this paper, we introduce and study a topology on qp . Spec g ( M ) {\mathop{\rm qp.Spec}\nolimits_{g}(M)} , called the quasi-Zariski topology, and investigate the properties of this topology and some conditions under which ( qp . Spec g ( M ) , q . τ g ) {(\mathop{\rm qp.Spec}\nolimits_{g}(M),q.\tau^{g})} is a Noetherian, spectral space.
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