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A torsion theory is called differential (higher differential) if a derivation(higher derivation) can be extended from any module to the module of quotientscorresponding to the torsion theory. We study conditions equivalent to higherdifferentiability of a torsion theory. It is known that the Lambek, Goldie andany perfect torsion theories are differential. We show that these classes oftorsion theories are higher differential as well. Then, we study conditionsunder which a higher derivation extended to a right module of quotients extendsalso to a right module of quotients with respect to a larger torsion theory.Lastly, we define and study the symmetric version of higher differentialtorsion theories. We prove that the symmetric versions of the results on higherdifferential (one-sided) torsion theories hold for higher derivations onsymmetric modules of quotients. In particular, we prove that the symmetricLambek, Goldie and any perfect torsion theories are higher differential.

Extending higher derivations to rings and modules of quotients