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

TL;DR

This paper generalizes the concept of higher derivations in torsion theories, showing they can be extended to modules of quotients and analyzing the invariance properties of these extensions.

Contribution

It introduces the notion of $(eta, eta)$-higher derivations, extending previous results to broader classes of torsion theories and modules.

Findings

Higher derivations can be extended from modules to their modules of quotients.

Extensions of higher derivations are consistent across larger torsion theories.

Some non-hereditary torsion theories are not differential.

Abstract

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.