Stable torsion theories and the injective hulls of simple modules. (Q2879413)

It is known that if a Noetherian ring \(R\) has the property that injective hulls of simple left \(R\)-modules are locally Artinian (Property (\(\diamond\))), then \(R\) satisfies Jacobson's conjecture, i.e. \(\bigcap_{i=1}^\infty J^i=0\), where \(J\) denotes the Jacobson radical \(J\) of \(R\). The paper under review links torsion theories and Property (\(\diamond\)). The main result of the paper shows that a left Noetherian ring \(R\) satisfies Property (\(\diamond\)) if and only if Dickson's torsion theory is stable.NEWLINENEWLINE Stability of torsion theories, in particular of Dickson's torsion theory, is then investigated further. It is shown that Dickson's torsion theory is stable if every cyclic singular left \(R\)-module has a nonzero socle. (Rings with this property have been called \(C\)-rings.) A few sufficient conditions for Noetherian rings to be \(C\)-rings are then found.NEWLINENEWLINE \textit{M. L. Teply} [Proc. Am. Math. Soc. 27, 29-34 (1971; Zbl 0211.36701)] considered a property (P) for a torsion theory \(\tau\), which means that every nonzero \(\tau\)-torsionfree module contains a nonzero projective submodule. It is shown that if \(R\) is a ring satisfying condition (P) for Dickson's torsion theory, then \(R\) is a \(C\)-ring. In fact, for a \(C\)-ring \(R\), a necessary and sufficient condition for Dickson's torsion theory to satisfy condition (P) is that every soclefree left ideal contains a nonzero projective left ideal.