\(^*\)-multilinear polynomials with invertible values
type: Item
instance of: scholarly article
title: \(^*\)-multilinear polynomials with invertible values
zbMATH Open document ID: 0805.16023
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1992-09-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108241
review text: If \(R\) is an algebra with involution \(*\) over the field \(F\), then a \(*\)- polynomial is any \(p(x_ 1,\dots,x_ n\), \(y_ 1,\dots,y_ n) \in F\{x_ i,y_ j\}\), the free algebra over \(F\), which is evaluated on \((r_ 1,\dots,r_ n) \in R^ n\) by substituting \(r_ i\) for \(x_ i\) and \(r_ i^*\) for \(y_ i\). A \(*\)-polynomial \(p(x_ 1,\dots,y_ n)\) is called multilinear if exactly one of \(x_ i\) or \(y_ i\) appears in each of its monomials. Let \(R^ o\) be the opposite ring of \(R\). The main result of the paper determines the structure of \(R\) when \(\text{char }F \neq 2\), \(\text{card }F > 5\), \(R\) is semi-prime, and \(p(x_ 1,\dots,y_ n)\) is a \(*\)-multilinear polynomial with every evaluation in \(R\) either zero or invertible, and with some evaluation invertible. In this case, there is a division ring \(D\) so that either \(R = M_ k(D)\) with \(\dim_{Z(D)}D\) finite when \(k \geq 2\) and the values of \(p\) are central when \(k \geq 3\), or else \(R = M_ k(D) \oplus M_ k(D)^ o\) with \((A,B^ o)^* = (B,A^ o)\) and if \(k \geq 2\) then \(\dim_{Z(D)}D\) is finite and the values of \(p\) are central.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 60479
zbMATH Keywords: invertible values
zbMATH Keywords: algebra with involution
zbMATH Keywords: free algebra
zbMATH Keywords: opposite ring
zbMATH Keywords: \(*\)-multilinear polynomial
zbMATH Keywords: division ring
author: Angela Valenti
author: M. Onofrio Di Vincenzo
MaRDI profile type: Publication
cites work: Multilinear polynomials with invertible values
cites work: Nill∗-polynomials in rings
cites work: Q4112071
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cites work: Nil and Power-Central Polynomials in Rings
cites work: Q3910639
description: scientific article; zbMATH DE number 60479