Unitary elements in simple Artinian rings
type: Item
instance of: scholarly article
title: Unitary elements in simple Artinian rings
zbMATH Open document ID: 0835.16027
author: Chen-Lian Chuang
author: P.-H. Lee
published in: Journal of Algebra
publication date: 1996-04-09T00:00:00Z
review text: Let \(R\) be a ring with involution *. An element \(u \in R\) is unitary if \(uu^*=1=u^*u\), and is a Cayley unitary if \(u=(1-k)(1+k)^{-1}\) for some \(k^*=-k\in R\). The authors characterize those unitary elements in simple Artinian rings which are products of Cayley unitaries. The first theorem shows that when \(R=M_n(D)\) for \(D\) a division ring, \(\text{char }D\neq 2\), and * is not the identity on \(D\cdot I_n\), then every unitary in \(R\) is a product of two Cayley unitary elements. If \(^*\) is the identity on \(D\cdot I_n\), then \(D=F\), a field, and * is said to be of the first kind. In this situation every Cayley unitary has determinant one. The second theorem is that if \(R=M_n(F)\) with \(\text{char }F \neq 2\) and * is of the first kind, then every unitary element in \(R\) of determinant one is a product of two Cayley unitary elements, except when \(R=M_2(\mathbb{F}_3)\) and \(A^*=\text{Diag}(1,- 1)A^t\text{Diag}(1,-1)\), where \(A^t\) is the usual transpose of \(A\) in \(R\). This exceptional case has \(I_2\) as the only Cayley unitary.
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 815919
zbMATH Keywords: rings with involution
zbMATH Keywords: unitary elements
zbMATH Keywords: simple Artinian rings
zbMATH Keywords: products of Cayley unitaries
zbMATH Keywords: Cayley unitary elements
reviewed by: Charles Lanski
MaRDI profile type: Publication
full work available at URL: jabr.1995.1253
OpenAlex ID: W2020679873
DOI: 10.1006/JABR.1995.1253
description: scientific article; zbMATH DE number 815919