Notes on generalized \((\sigma,\tau)\)-derivation.
type: Item
instance of: scholarly article
title: Notes on generalized \((\sigma,\tau)\)-derivation.
zbMATH Open document ID: 1202.16036
DOI: 10.4171/RSMUP/123-6
author: Emine Koç
author: Öznur Gölbaşı
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2010-09-07T00:00:00Z
full work available at URL: https://eudml.org/doc/240502
full work available at URL: http://rendiconti.math.unipd.it/volumes/vol123.php
review text: Let \(R\) be a prime ring with automorphisms \(\sigma\) and \(\tau\), \((\sigma,\tau)\)-derivation \(d\), and generalized \((\sigma,\tau)\)-derivation \((f,d)\). The main results in the paper claim that when \(f\) acts as a homomorphism, or as an antihomomorphism, then \(d=0\), and that for \(f\) acting as a homomorphism on a Lie ideal \(U\) of \(R\), then either \(U\) is central or \(d=0\). Theorem 2, when \(f\) is a homomorphism, is incorrect as stated, so leads to an incorrect statement for Theorem 3 about antihomomorphisms, and the same problem holds for Theorem 5, which is Theorem 2 for Lie ideals. The error in the proof of these results is the authors' assumption that \(\tau-\sigma\) is a homomorphism, which cannot be true unless \(\tau=\sigma\). The authors fail to realize that for any \(\sigma\) and \(\tau\), \(d=\tau-\sigma\) is a \((\sigma,\tau)\)-derivation and \((\tau,\tau-\sigma)\) is a generalized \((\sigma,\tau)\)-derivation with \(\tau\) a homomorphism. Finally, the proof of Theorem 6 requires the assumption that \(U\) is closed under squares, so contains \(2uv\) for \(u,v\in U\).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 5780247
zbMATH Keywords: prime rings
zbMATH Keywords: automorphisms
zbMATH Keywords: additive mappings
zbMATH Keywords: generalized derivations
zbMATH Keywords: Lie ideals
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5780247