A Note on (α, β)-higher Derivations and their Extensions to Modules of Quotients

Abstract

We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of (α, β)-derivation to (α, β)-higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for α and β is (α, β)-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. Vaš, 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.