Zariski spaces of modules (Q2376568)

Let \(R\) be a ring and \(M\) be an \(R\)-module. Suppose that \(\mathcal{L}(M)\) denote the set of all submodules of \(M\), \(\mathcal{X}\subset \mathcal{L}(R)\setminus\{R\}\) and \(\mathcal{F}\subset \mathcal{L}(M)\setminus\{M\}\). For any submodule \(N\) of \(M\) put \(\mathcal{F}V(N)=\{K\in \mathcal{F}: N\subset K\}\) (the \(\mathcal{F}\)-variety of \(N\)) and \(\mathcal{F}\zeta(M)=\{\mathcal{F}V(N)\): \(N\) is a submodule of \(M\}\). If \(\mathcal{X}\zeta(R)\) (i.e., the collection of all \(\mathcal{X}\)-varieties of all ideals of \(R\)) is a topology on \(\mathcal{X}\), we say that \((\mathcal{X}, \mathcal{X}\zeta(R))\) is the Zariski topology of \(R\) and \(\mathcal{X}\) determine a Zariski topology of \(R\), in this case, we call the semiring \((\mathcal{X}\zeta(R), \cap, \cup)\) the \(\mathcal{X}\)-Zariski semiring of \(R\). Suppose that \(\mathcal{F}\zeta(M)\) is a semimodule over the \(\mathcal{X}\)-Zariski semiring of \(R\) with the operations \[ \mathcal{F}V(N)+\mathcal{F}V(N^{\prime})=\mathcal{F}V(N)\cap\mathcal{F}V(N^{\prime} \text{ and }\mathcal{X}V(I)\mathcal{F}{N}=\mathcal{F}V(IM)\cup\mathcal{F}V(N), \] where \(N\) and \(N^{\prime}\) are submodules of \(M\) and \(I\) is an ideal of \(R\). Then we call \((\mathcal{F}, \mathcal{F}\zeta(M))\) the \(\mathcal{F}\)-Zariski space of \(M\) and we say that \(\mathcal{F}\) determine a Zariski space of \(M\). Also, in this case, \(\mathcal{F}\zeta(M)\) is called the \(\mathcal{F}\)-Zariski semimodules of \(M\) over \(\mathcal{X}\zeta(R)\). This paper deals with the fundamental properties of Zariski spaces of a module which has been introduced by \textit{A. Nikseresht} and \textit{A. Azizi} [J. Pure Appl. Algebra 217, No. 7, 1187--1194 (2013; Zbl 1273.13019)]. This class of spaces is a generalization of Zariski topology on prime submodules. After giving a clear and informative introduction, the authors show that whenever \(\mathcal{X}\) determine a Zariski topology of \(R\), then \(\mathcal{F}\zeta(M)\) is a Zariski semimodule over \(\mathcal{X}\zeta(R)\) iff \((\mathcal{F}: M)\subset \mathcal{X}RAD(R)\) (see Theorem \(2.7\)). Note that \((\mathcal{F}: M)=\{(F: M)| F\in\mathcal{F}\}\), where \((F:M)=\{r\in R: rM\subset F\}\) and for submodule \(N\) of \(M\) \(\mathcal{F}\mathrm{-rad}(N)=\bigcap_{K\in\mathcal{F}V(N)}K\) is the \(\mathcal{F}\)-radical of \(N\) in \(M\). In Theorem \(2.9\), they show that if \(\mathcal{F}\) and \(\mathcal{F^{\prime}}\) are Zariski spaces of \(R\)-modules \(M\) and \(M^{\prime}\) over \(\mathcal{X}\zeta(R)\) respectively, \(\phi:\mathcal{F}\zeta(M)\longrightarrow\mathcal{F^{\prime}}\zeta(M^{\prime})\) is an \(\mathcal{X}\zeta(R)\)-isomorphism and \(\mathcal{F}\zeta(M)\) is a Zariski semimodule over a Zariski semiring \(\mathcal{X^{\prime}}\zeta(R)\) of \(R\), then \(\mathcal{F^{\prime}}\zeta(M)\) is also a Zariski \(\mathcal{X^{\prime}}\zeta(R)\)-semimodule and \(\phi\) is an \(\mathcal{X^{\prime}}\zeta(R)\)-isomorphism. Indeed, in section \(3\), the well known results of the prime Zariski space of \(M\) are proved for every Zariski space.