Centralizing maps in prime rings with involution
type: Item
instance of: scholarly article
title: Centralizing maps in prime rings with involution
zbMATH Open document ID: 0815.16016
author: Matej Brešar
author: C. R. Miers
author: W. S. III Martindale
published in: Journal of Algebra
publication date: 1995-05-15T00:00:00Z
full work available at URL: http://hdl.handle.net/1828/2661
review text: Let \(R\) be a ring with center \(Z\). For \(x, y \in R\), denote \([x,y] = xy - yx\). An additive map \(f : K \to K\) defined on a subset \(K \subset R\) is called centralizing if \([f(x), x] \in Z\) for all \(x\in K\). Centralizing maps have been studied by a number of authors, starting with Posner in 1957. The present paper focuses on the case where \(R\) is a prime ring of characteristic different from 2 and \(K\) is the group of skew-symmetric elements in \(R\) under some involution \(*\). The main result is a description of all the centralizing maps on \(K\), under the hypotheses that \([K,K] \neq \{0\}\) and that \(R\) is not an order in a 16-dimensional central simple algebra. It is shown that every such map \(f\) has the form \(f(x) = \lambda x + \mu(x)\) for some \(\lambda\) in the extended centroid \(C\) of \(R\) and some additive map \(\mu : K \to C\). In particular, the map \(f\) is actually commuting, in the sense that \([f(x),x] = 0\) for all \(x \in K\).
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16U70
zbMATH DE Number: 554246
zbMATH Keywords: additive map
zbMATH Keywords: prime ring
zbMATH Keywords: skew-symmetric elements
zbMATH Keywords: involution
zbMATH Keywords: centralizing maps
zbMATH Keywords: extended centroid
reviewed by: Jean-Pierre Tignol
MaRDI profile type: Publication
OpenAlex ID: W2006310784
DOI: 10.1006/JABR.1993.1223
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description: scientific article; zbMATH DE number 554246