$(\sigma,\tau)$-Derivations of Group Rings

TL;DR

This paper investigates $(\sigma, au)$-derivations of group rings over integral domains, extending known results for derivations on integral group rings and generalizing Skolem-Noether applications to this context.

Contribution

It introduces the study of $(\sigma, au)$-derivations in group rings and extends classical derivation results to this broader setting, including a generalization of Skolem-Noether theorem applications.

Findings

Extended derivation results to $(\sigma, au)$-derivations on $Z G$

Generalized Skolem-Noether theorem for $(\sigma, au)$-derivations

Provided conditions on $\sigma$ and $ au$ for finite groups

Abstract

We study $(\sigma,\tau)$-derivations of a group ring $RG$ of a finite group $G$ over an integral domain $R$ with $1$. As an application we extend a well known result on derivation of an integral group ring $\Bbb{Z}G$ to $(\sigma,\tau)$-derivation on it for a finite group $G$ with some conditions on $\sigma$ and $\tau$. In the process of the extension, a generalization of an application of Skolem-Noether Theorem to derivation on a finite dimensional central simple algebra has also been given for the $(\sigma,\tau)$-derivation case.