On the identity \(h(x)=af(x)+g(x)b\).
type: Item
instance of: scholarly article
title: On the identity \(h(x)=af(x)+g(x)b\).
zbMATH Open document ID: 1048.16018
author: Jui-Chi Chang
published in: Taiwanese Journal of Mathematics
publication date: 2003-01-01T00:00:00Z
review text: Let \(\alpha\) and \(\beta\) be automorphisms of a ring \(R\), and let \(\delta\) be an \((\alpha,\beta)\)-derivation of \(R\). The author defines the concept of a generalized \((\alpha,\beta)\)-derivation of \(R\) associated with \(\delta\) as an additive map \(f\colon R\to R\) satisfying \[ f(xy)=f(x)\alpha(y)+\beta(x)\delta(y)=\delta(x)\alpha(y)+\beta(x)f(y) \text{ for all }x,y\in R. \] The main theorem describes \((\alpha,\beta)\)-derivations \(f\), \(g\) and \(h\) of a prime ring \(R\) such that \(h(x)=af(x)+g(x)b\) for all \(x\in R\), where \(a\) and \(b\) are fixed elements. This result generalizes a result by the reviewer [J. Algebra 156, No. 2, 385-394 (1993; Zbl 0773.16017)] who treated this condition for ordinary derivations. In corollaries some other results that are known for derivations are extended to generalized \((\alpha,\beta)\)-derivations.
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 1926032
zbMATH Keywords: generalized \((\alpha,\beta)\)-derivations
zbMATH Keywords: prime rings
zbMATH Keywords: additive maps
reviewed by: Matej Brešar
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407520
OpenAlex ID: W1874815768
DOI: 10.11650/TWJM/1500407520
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description: scientific article; zbMATH DE number 1926032