A note on $(\alpha ,\beta )$-higher derivations and their extensions to modules of quotients

We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of $(\alpha ,\beta )$-derivation to $(\alpha ,\beta )$-higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for $\alpha$ and $\beta$ is $(\alpha ,\beta )$-higher derivation invariant. As a consequence, any higher derivation can be extended from a module to its module of quotients. Then, we show that any higher derivation extended to a module of quotients extends also to a module of quotients with respect to a larger torsion theory in such a way that these extensions agree. We also demonstrate these results hold for symmetric filters as well. We finish the paper with answers to two questions posed in [L. Va\s, Extending higher derivations to rings and modules of quotients, International Journal of Algebra, 2 (15) (2008), 711–731]. In particular, we present an example of a non-hereditary torsion theory that is not differential.