Extending ring derivations to right and symmetric rings and modules of quotients

TL;DR

This paper introduces the symmetric version of differential torsion theories, extending derivation results to symmetric modules of quotients and analyzing their behavior in various ring contexts.

Contribution

It defines symmetric differential torsion theories and proves their properties, extending classical derivation results to symmetric modules of quotients and various ring of quotients.

Findings

Symmetric torsion theories are shown to be differential.

Derivations extend to larger torsion theories under certain conditions.

Results apply to maximal, total, and perfect rings of quotients.

Abstract

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total and perfect right and symmetric rings of quotients.