Extending higher derivations to rings and modules of quotients

TL;DR

This paper investigates conditions under which higher derivations can be extended to modules of quotients in torsion theories, establishing that well-known classes like Lambek, Goldie, and perfect torsion theories are higher differential.

Contribution

It extends the concept of differential torsion theories to higher derivations and introduces symmetric higher differential torsion theories, proving their properties for key classes.

Findings

Lambek, Goldie, and perfect torsion theories are higher differential.

Higher derivations extend to larger torsion theories under certain conditions.

Symmetric higher differential torsion theories behave similarly to one-sided cases.

Abstract

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