Differential torsion theory

Differential torsion theories are introduced and it is shown that for a hereditary torsion theory ττ every derivation on an R-module M   has a unique extension to its module of quotients if and only if ττ is a differential torsion theory. Dually, we show that when ττ is cohereditary, every derivation on M can be lifed uniquely to its module of coquotients.