An independence property of central polynomials
type: Item
instance of: scholarly article
title: An independence property of central polynomials
zbMATH Open document ID: 0760.16007
author: Chen-Lian Chuang
published in: Pacific Journal of Mathematics
publication date: 1992-06-25T00:00:00Z
review text: Let \(\Phi\) be a field and \(X=\{x_ 1,x_ 2,\dots,x_ n,\dots\}\) be a set of noncommuting indeterminates over \(\Phi\). Let \(\Phi\{X\}\) be the free algebra generated by \(X\). Theorem. Let \(I_ n=\{f\in\Phi\{X\}\): \(f\) is a polynomial identity of \(M_ n(\Phi)\}\) be the \(T\)-ideal of polynomial identities of \(M_ n(\Phi)\). Let \(f_ i(x_ 1,\dots,x_ r)\) and \(g_ i(x_{r+1},\dots,x_ s)\), \(1\leq i\leq k\) be polynomials in \(\Phi\{X\}\) with disjoint indeterminates. Assume that \(\sum_{i=1}^ k f_ i(x_ 1,\dots,x_ r)g_ i(x_{r+1},\dots,x_ s)\) is a central polynomial on \(M_ n(\Phi)\). Then except when \(k\geq 2\), \(n=2\) and \(\Phi=\mathbb{Z}/(2)\) the following hold: (1) If \(f_ i\), \(i=1,2,\dots,k\) are \(\Phi\)-independent modulo \(I_ n\) then \(g_ i\) must be central \(\forall i\). (2) If \(g_ i\), \(i=1,2,\dots,k\) are \(\Phi\)-independent modulo \(I_ n\) then \(f_ i\) must be central \(\forall i\). This generalizes a theorem of \textit{A. Regev} [Pac. J. Math. 83, 269-271 (1979; Zbl 0425.16017)]. Examples are given to show that the result is false when \(k\geq 2\), \(n=2\) and \(\Phi=\mathbb{Z}/(2)\).
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 11034
zbMATH Keywords: noncommuting indeterminates
zbMATH Keywords: free algebra
zbMATH Keywords: \(T\)-ideal
zbMATH Keywords: polynomial identities
zbMATH Keywords: central polynomial
reviewed by: Ram Niwas Gupta
MaRDI profile type: Publication
full work available at URL: pjm.1991.148.237
OpenAlex ID: W2031848741
DOI: 10.2140/PJM.1991.148.237
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description: scientific article; zbMATH DE number 11034