Second spectrum of modules and spectral spaces. (English) Zbl 1408.13021
All rings \(R\) in this paper are commutative with identity and all modules are unital right \(R\)-modules. Recall that a module \(M\) is said to be a second module if \(M \neq 0\) and \(\mathrm{ann}_{R}(M) =\mathrm{Ann}_{R}(M/N)\) for every proper submodule \(N\) of \(M\). A second submodule \(N\) of a module \(M\) is called a (\(P\)-)second submodule of \(M\) if it is a second module by itself (and will then have an annihilator that is a prime ideal \(P\) of \(R\). The set of all second modules of a module \(M\) is called the second spectrum of \(M\) and is denoted by \(\mathrm{Spec}^{s}(M)\). It is assumed that \(\mathrm{Spec}^{s}(M) \neq \phi\). The map \(\psi^{s} : \mathrm{Spec}^{s}(M) \rightarrow \mathrm{Spec}(R/\mathrm{Ann}_{R}(M))\) defined by \(\psi_{s}(S) = \mathrm{ann}_{R}(S)\) is called the natural map of \(\mathrm{Spec}^{s}(M)\). If this natural map is injective for an \(R\)-module \(M\), then \(M\) is said to be \(X^{s}\)-injective. \(X^{s}\)-injective modules are studied, a characterization is found of the second submodules of an \(X^{s}\)-injective module and conditions for the existence of a maximal second submodule are investigated. \(X^{s}\)-injectives over a one-dimensional domain are characterized.
Every weak co-multiplication module over a one-dimensional domain is shown to be cotop. These results are used to study the dual Zariski topology \((\mathrm{Spec}^{s}(M), \tau^{s})\). It is shown that if \(R\) is a Dedekind domain and \(M\) an \(X^{s}\)-injective \(R\)-module, then \((\mathrm{Spec}^{s}(M), \tau^{s})\) is a spectral space if and only if \(\mathrm{Spec}^{s}(M)\) is a finite set or \(\operatorname{div}(M_{R}) \neq 0\). More results on \((\mathrm{Spec}^{s}(M), \tau^{s})\) are found for specific rings, such as one-dimensional integral domains.
A module is said to be secondful if \(\psi^{s}\) is surjective. For a secondful module \(M\), the combinatorial dimension of \((\mathrm{Spec}^{s}(M), \tau^{s})\) and the Krull dimension of \(R/\mathrm{Ann}_{R}(M)\) are shown to be equal and several equivalences are found for the combinatorial dimension of \((\mathrm{Spec}^{s}(M), \tau^{s})\) to be zero. In this case, all the irreducible components of \((\mathrm{Spec}^{s}(M), \tau^{s})\) are found.
The final section deals with conditions for \(\mathrm{Spec}^{s}(M)\) to be a spectral space in the case of two different topologies.
Every weak co-multiplication module over a one-dimensional domain is shown to be cotop. These results are used to study the dual Zariski topology \((\mathrm{Spec}^{s}(M), \tau^{s})\). It is shown that if \(R\) is a Dedekind domain and \(M\) an \(X^{s}\)-injective \(R\)-module, then \((\mathrm{Spec}^{s}(M), \tau^{s})\) is a spectral space if and only if \(\mathrm{Spec}^{s}(M)\) is a finite set or \(\operatorname{div}(M_{R}) \neq 0\). More results on \((\mathrm{Spec}^{s}(M), \tau^{s})\) are found for specific rings, such as one-dimensional integral domains.
A module is said to be secondful if \(\psi^{s}\) is surjective. For a secondful module \(M\), the combinatorial dimension of \((\mathrm{Spec}^{s}(M), \tau^{s})\) and the Krull dimension of \(R/\mathrm{Ann}_{R}(M)\) are shown to be equal and several equivalences are found for the combinatorial dimension of \((\mathrm{Spec}^{s}(M), \tau^{s})\) to be zero. In this case, all the irreducible components of \((\mathrm{Spec}^{s}(M), \tau^{s})\) are found.
The final section deals with conditions for \(\mathrm{Spec}^{s}(M)\) to be a spectral space in the case of two different topologies.
Reviewer: Frieda Theron (Potchefstroom)
MSC:
| 13C13 | Other special types of modules and ideals in commutative rings |
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
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