On the nilindex of the radical of a relatively free associative algebra.
type: Item
instance of: scholarly article
title: On the nilindex of the radical of a relatively free associative algebra.
zbMATH Open document ID: 1153.16021
DOI: 10.1134/S0001434607090283
author: L. M. Samoilov
published in: Mathematical Notes
publication date: 2008-09-03T00:00:00Z
review text: A classical result of Amitsur states that the Jacobson radical of a relatively free associative algebra over an infinite field is nil; Kemer showed that each finitely generated T-ideal contained in the Jacobson radical of a relatively free algebra is nil of bounded index. Let \(F\) be an infinite field of positive characteristic \(p\) and let \(F(X)\) be the free associative algebra on an infinite set of variables \(X\). If \(\Gamma\) is a T-ideal in \(F(X)\) then the complexity of \(\Gamma\) equals the largest positive integer \(n\) such that the T-ideal of the full matrix algebra \(M_n(F)\) contains \(\Gamma\). The main theorem of the paper under review states that if the complexity \(n\) of \(\Gamma\) satisfies \(n<p\) then the Jacobson radical of \(F(X)/\Gamma\) is nil of bounded index. In the course of proof the author proves the following result that is quite interesting. If \(n<p\) then every trace identity for \(M_n(F)\) follows from the identitites \(T(1)=n\) and from the complete linearization of the Cayley-Hamilton identity. Here \(T(a)\) stands for the trace of the matrix \(a\).
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 5319866
zbMATH Keywords: nil ideals
zbMATH Keywords: relatively free associative algebras
zbMATH Keywords: nil-indices
zbMATH Keywords: radicals
zbMATH Keywords: bases for trace identities
zbMATH Keywords: Cayley-Hamilton identity
MaRDI profile type: Publication
cites work: Q4370153
cites work: Basis of the identities of the matrix algebra of order two over a field of characteristic \(p\neq 2\)
cites work: Identities of associative algebras
cites work: Varieties and algebraic algebras of bounded degree
cites work: MULTILINEAR IDENTITIES OF THE ALGEBRAS OVER A FIELD OF CHARACTERISTIC P
cites work: Q5677619
full work available at URL: s0001434607090283
OpenAlex ID: W2053481108
description: scientific article; zbMATH DE number 5319866