Torsion classes generated by silting modules

Abstract

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion $\mathcal{T}$ such that the regular module has a special $\mathcal{T}$-preenvelope. In particular, every torsion-enveloping class in $\mathrm{Mod}\textrm{-}R$ are of the form $\mathrm{Gen}(T)$ for a minimal silting module $T$. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form $\mathrm{Cogen}(T)$, where $T$ is a cosilting module.