Topologies on \(\mathrm{Spec}_{g}(M)\) (Q2895937)

In this review \(R\) denotes a commutative ring with identity. Let \(G\) be an arbitrary group. The ring \(R\) is \textit{\(G\)-graded} if it has a direct sum decomposition \(R=\bigoplus_{g\in G} R_g\) such that for all \(g,h\in G\), \(R_gR_h\subseteq R_{gh}\). Then an \(R\)-module \(M\) is said to be a \textit{graded module} over the graded ring \(R\) if it has a direct sum decomposition \(M=\bigoplus_{g\in G} M_g\) such that for all \(g,h\in G\), \(R_gM_h\subseteq M_{gh}\). If \(N\) is a proper graded submodule of \(M\), then \(N\) is a \textit{prime graded submodule} if for any homogeneous elements \(a\in R\) and \(x\in M\) either \(x\in N\) or \(a\in (N:M)\). Let \(\text{Spec}_g(M)\) be the set of all prime graded submodules of \(M\). For a subset \(E\) of \(M\) define NEWLINE\[NEWLINEV_\ast^g(E)=\{P\in \text{Spec}_g(M)\,|\,E\subseteq P\}.NEWLINE\]NEWLINE Now put NEWLINE\[NEWLINE\zeta_\ast^g(M)=\{V_\ast^g(N)\,|\,N \text{ is graded submodule of }M\}.NEWLINE\]NEWLINE Then \(\zeta_\ast^g(M)\) satisfies the axioms for the closed sets of a unique topology \(\tau_\ast^g\) on \(\text{Spec}_g(M)\), except not necessarily being closed under finite unions. We say that \(M\) is a \textit{\(g\)-Top module} if \(\zeta_\ast^g(M)\) is closed under finite unions. If \(M\) is a \(g\)-Top module, then the unique topology \(\tau_\ast^g\) on \(\text{Spec}_g(M)\), having \(\zeta_\ast^g(M)\) as the set of closed sets, is called the \textit{quasi-Zariski topology}. In the paper under review the author introduces this topology and proved various properties of \(g\)-Top modules and this topology on their graded spectrum. He showed that the set \(\{X_f\,|\,f \text{ homogeneous element of }M\}\), where \(X_f=X\setminus V_\ast^g(f)\), is a basis of the quasi-Zariski topology.NEWLINENEWLINEThe author has also introduced another topology on \(\text{Spec}_g(M)\) and called it the \textit{Zariski topology}. It is defined in the following way. For every graded submodule \(N\) of \(M\) define NEWLINE\[NEWLINEV^g(N)=\{P\in \text{Spec}_g(M)\,|\, (P:M)\supseteq (N:M)\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\zeta^g(M)=\{V^g(N)\,|\,N \text{ is a graded submodule of }M\}.NEWLINE\]NEWLINE The author proves that for any graded module \(M\) there always exists a topology \(\tau^g\) on \(\text{Spec}_g(M)\) in which \(\zeta^g(M)\) is the family of closed sets and showed some properties of this topology.