Differentiability of torsion theories

We prove that every perfect torsion theory for a ring R is differential (in the sense of [P.E. Bland, Differential torsion theory, Journal of Pure and Applied Algebra 204 (2006) 1–8]). In this case, we construct the extension of a derivation of a right R-module M to a derivation of the module of quotients of M. Then, we prove that the Lambek and Goldie torsion theories for any R are differential.