$f$-derivations on rings and modules

summary:If $\tau $ is a hereditary torsion theory on $\bold{Mod}_{R}$ and $Q_{\tau }:\bold{Mod}_{R}\rightarrow \bold{Mod}_{R}$ is the localization functor, then we show that every $f$-derivation $d:M\rightarrow N$ has a unique extension to an $f_{\tau }$-derivation $d_{\tau }:Q_{\tau }(M)\rightarrow Q_{\tau }(N)$ when $\tau $ is a differential torsion theory on $\bold{Mod}_{R}$. Dually, it is shown that if $\tau $ is cohereditary and $C_{\tau }:\bold{Mod}_{R}\rightarrow \bold{Mod}_{R}$ is the colocalization functor, then every $f$-derivation $d:M\rightarrow N$ can be lifted uniquely to an $f_{\tau }$-derivation $d_{\tau }:C_{\tau }(M)\rightarrow C_{\tau }(N)$