New polynomial identities for \(2\times 2\) generic matrices in characteristic \(2\)
type: Item
instance of: scholarly article
title: New polynomial identities for \(2\times 2\) generic matrices in characteristic \(2\)
zbMATH Open document ID: 0935.16010
DOI: 10.1016/S0022-4049(97)00185-0
author: Vesselin Drensky
author: Dimitar Tsiganchev
published in: Journal of Pure and Applied Algebra
publication date: 2000-02-08T00:00:00Z
review text: For a commutative ring \(K\) let \(R_m(K)\) denote the \(K\)-algebra generated by \(m\) generic \(2\times 2\) matrices. The kernel of the canonical homomorphism \(\vartheta_m\colon R_m(\mathbb{Z})\to R_m(\mathbb{Z}/2\mathbb{Z})\) is studied, where \(\mathbb{Z}\) denotes the ring of integers. Answering Procesi's question \textit{W. F. Schelter} [J. Algebra 96, 48-53 (1985; Zbl 0577.16009)] has shown the existence of an element in the kernel of \(\vartheta_5\) which is not contained in \(2R_5(\mathbb{Z})\). This \(5\)-variable element is multihomogeneous of degree \(6\). In the present paper it is proved that there exists a multilinear element of degree \(5\) in \(\text{Ker}(\vartheta_5)\setminus 2R_5(\mathbb{Z})\). This result is obtained by comparing the dimensions of the multilinear polynomial identities of degree \(5\) for the \(2\times 2\) matrix algebras over the rationals and \(\mathbb{Z}/2\mathbb{Z}\).
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R30
zbMATH DE Number: 1308270
zbMATH Keywords: generic matrices
zbMATH Keywords: polynomial identities
zbMATH Keywords: trace rings
reviewed by: Matyas Domokos
MaRDI profile type: Publication
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full work available at URL: s0022-4049(97)00185-0
OpenAlex ID: W2011477332
description: scientific article; zbMATH DE number 1308270