Gabriel topologies on Dubrovin valuation rings (Q2777586)

An order \(R\) in a simple Artinian ring \(Q\) is called by the authors a Dubrovin valuation ring if for any \(q\in Q\setminus R\) there exist \(r,r_1\in R\) such that \(qr,r_1q\in R\setminus J(R)\) and \(R/J(R)\) is local. In such a ring \(R\), if \(Q\) has finite dimension over its center, all right Gabriel topologies (equivalently: all hereditary torsion theories of Mod-\(R\)) are described in terms of prime ideals of \(R\) and indecomposable injective cogenerators.NEWLINENEWLINENEWLINELet \(P\) be a prime ideal of \(R\) (\(P\in\text{Spec}(R)\)) and \({\mathcal C}(P)=\{c\in R\mid c\) is regular mod \(P\}\). Then two right Gabriel topologies are defined as follows: \({\mathcal T}_P=\{I\mid I\) is a right ideal with \(I\cap{\mathcal C}(P)\neq\emptyset\}\), \(\overline{\mathcal T}_P=\{I\mid I\) is a right ideal with \(I\supseteq P\}\) if \(P\neq P^2\).NEWLINENEWLINENEWLINEThe main result: for every right Gabriel topology \(\mathcal T\) of \(R\) there exists \(P\in\text{Spec}(R)\) such that either \({\mathcal T}={\mathcal T}_P\) or \({\mathcal T}=\overline{\mathcal T}_P\) and \(P=P^2\) (in this case the lattice Tors-\(R\) of all hereditary torsion theories of Mod-\(R\) is a chain).NEWLINENEWLINENEWLINEA right Gabriel topology \(\mathcal T\) is cogenerated by an injective module \(E\) if \({\mathcal T}=\{I\mid I\) is a right ideal of \(R\) with \(\Hom_R(R/I,E)=0\}\). All the right Gabriel topologies on a Dubrovin valuation ring \(R\) are cogenerated by indecomposable injective right \(R\)-modules \(E\) of two types: (1) \(\mathcal T_P\) is cogenerated by \(E\) such that \(P=\text{Ass}_R(E)\), and there exists \(0\neq x\in E\) with \(xP=0\), where \(\text{Ass}_R(E)=\{r\in R\mid Nr=0\) for some nonzero submodule \(N\subseteq E\}\); (2) if \(P^2=P\) then \(\overline{\mathcal T}_P\) is cogenerated by \(E\) such that \(P=\text{Ass}_R(E)\) and \(xP\neq 0\) for any \(0\neq x\in E\).