A note on ${(\sigma,\tau)}$-Derivations on Commutative Algebras

TL;DR

This paper investigates the properties of $(\sigma, au)$-derivations in commutative algebras, characterizing them over quadratic number fields and extending known results from UFDs to non-UFDs.

Contribution

It provides a characterization of $(\sigma, au)$-derivations over rings of integers in quadratic fields and extends classical results to non-UFD contexts.

Findings

Characterization of $(\sigma, au)$-derivations over quadratic integer rings

Extension of derivation results from UFDs to certain non-UFDs

Analysis of universal mapping properties of these derivations

Abstract

We study universal mapping properties of $(\sigma,\tau)$-derivations over commutative algebras and characterize them over rings of integers of quadratic number fields. As a result we provide extension of some well known results on UFD's of such derivations to certain non-UFD's as well.