## Paul E. Bland

*$f$-derivations on rings and modules *

Comment.Math.Univ.Carolin. 47,3 (2006) 379-390. **Abstract:**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 $d_{\tau }:C_{\tau }(M)\rightarrow C_{\tau }(N)$.

**Keywords:** torsion theory, differential filter, localization, colocalization, $f$-derivation

**AMS Subject Classification:** Primary 16S90, 16W25; Secondary 16D99

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