Minimal polynomials of algebraic derivations and automorphisms.
type: Item
instance of: scholarly article
title: Minimal polynomials of algebraic derivations and automorphisms.
zbMATH Open document ID: 1122.16029
author: Chen-Lian Chuang
author: Tsiu-Kwen Lee
author: Chi-Tsuen Yeh
published in: Linear Algebra and its Applications
publication date: 2007-05-10T00:00:00Z
review text: Let \(R\) be a prime ring with extended centroid \(C\), and let \(a\in R\) be \(C\)-algebraic. The paper describes the minimal polynomials of the inner derivation \(x\mapsto[a,x]\) and (for an invertible \(a\)) the inner automorphism \(x\mapsto axa^{-1}\) in terms of the minimal polynomial of \(a\). The results are definitive of their kind, but somewhat too technical to be stated here. They are in fact derived from more general results on minimal polynomials of maps \(x\mapsto ax+xb\) and \(x\mapsto axb\) where \(a,b\) are \(C\)-algebraic elements in the symmetric Martindale ring of quotients of \(R\). As a corollary it is shown that the minimal polynomial of a \(C\)-algebraic derivation (resp. automorphism) takes the form \(\lambda^lh(\lambda)\) (resp. \((\lambda-1)^lh(\lambda)\)), where \(h(\lambda)\in C[\lambda]\) is such that \(h(0)\neq 0\) (resp. \(h(1)\neq 0\)) and \(l\) is odd if \(\text{char}(R)\neq 2\); if, however, \(\text{char}(R)=2\), then \(l\) is a \(2\)-power.
reviewed by: Matej Brešar
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 5151572
zbMATH Keywords: minimal polynomials
zbMATH Keywords: algebraic derivations
zbMATH Keywords: algebraic automorphisms
zbMATH Keywords: prime rings
Wikidata QID: Q57788098
MaRDI profile type: Publication
full work available at URL: j.laa.2007.01.007
OpenAlex ID: W2052630750
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DOI: 10.1016/J.LAA.2007.01.007
description: scientific article; zbMATH DE number 5151572