Identities with skew derivations
type: Item
instance of: scholarly article
title: Identities with skew derivations
zbMATH Open document ID: 0952.16025
author: Chen-Lian Chuang
published in: Journal of Algebra
publication date: 2001-01-09T00:00:00Z
review text: In this long and complex but well written paper the author extends the seminal work of \textit{V. K. Kharchenko} [Algebra Logika 17, 220-238 (1978; Zbl 0423.16011), and ibid. 18, 86-119 (1979; Zbl 0464.16027)] to skew derivations of prime rings \(R\). Roughly speaking a differential polynomial is a generalized polynomial with variables \(x^H_j\) for \(H\) a composition of derivations (and (anti) automorphisms) of \(R\). Substitution of \(r_j\in R\) for \(x^H_j\) yields \(H(r_j)\). A differential identity for \(R\) is a differential polynomial so that all evaluations result in zero. The approach of the author extends the work of Kharchenko by allowing exponents of the variables to be compositions including skew derivations and more generally \((g,h)\) derivations or higher derivations. When \(g\) and \(h\) are automorphisms of \(R\) a \((g,h)\) derivation \(D\) is an additive endomorphism of \(R\) so that \(D(ab)=g(a)D(b)+ D(a)h(b)\). The aim of the paper is to show that when \(R\) satisfies an identity with these general exponents then \(R\) satisfies a generalized polynomial identity. Much of the paper concerns the definition and identification of the trivial identities, essentially those which hold by the definitions of the maps under consideration. A very simple example of a trivial identity would be \(x^E-x^{DF}+x^{FD}\) where \(D\) and \(F\) are derivations of \(R\) and \(E\) is the derivation \(DF-FD\).
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 1428244
zbMATH Keywords: skew derivations of prime rings
zbMATH Keywords: differential polynomials
zbMATH Keywords: differential identities
zbMATH Keywords: higher derivations
zbMATH Keywords: automorphisms
zbMATH Keywords: additive endomorphisms
zbMATH Keywords: generalized polynomial identities
zbMATH Keywords: trivial identities
reviewed by: Charles Lanski
MaRDI profile type: Publication
full work available at URL: jabr.1999.8052
OpenAlex ID: W2016053219
cites work: Q4865491
cites work: ∗-Differential Identities of Prime Rings with Involution
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cites work: Q4002121
cites work: Skew derivations of prime rings
cites work: Q4948659
cites work: Q5519166
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DOI: 10.1006/JABR.1999.8052
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description: scientific article; zbMATH DE number 1428244