All hereditary torsion theories are differential. (Q1004458)

Let \(R\) be an associative ring endowed with a derivation map \(\delta\) and \(\tau\) a hereditary torsion theory on the category Mod-\(R\) of right \(R\)-modules, with associated Gabriel filter \(\mathcal F_\tau\) and associated torsion radical \(t_\tau\). Sharpening an earlier result of Golan, \textit{P. E. Bland} [J. Pure Appl. Algebra 204, No. 1, 1-8 (2006; Zbl 1102.16020)] proved that the Gabriel filter \(\mathcal F_\tau\) is \(\delta\)-invariant (in the sense that for every \(I\in\mathcal F_\tau\) there exists \(J\in\mathcal F_\tau\) such that \(\delta[J]\subseteq I\)) if and only if, for every \(M\in\text{Mod-}R\) and \(\delta\)-derivation \(d\) on \(M\), \(d[t_\tau(M)]\subseteq t_\tau(M)\). Bland called the hereditary torsion theories with this property differential and \textit{L. Vaš} [J. Pure Appl. Algebra 210, No. 3, 847-853 (2007; Zbl 1117.16021)] asked whether all hereditary torsion theories are differential. In this paper the authors answer this question in the affirmative by proving a slightly more general result on skew-derivations.