On the extension of automorphisms of polynomial rings
type: Item
instance of: scholarly article
title: On the extension of automorphisms of polynomial rings
zbMATH Open document ID: 0866.16023
DOI: 10.1007/BF02107336
published in: Siberian Mathematical Journal
publication date: 1997-05-29T00:00:00Z
review text: Let \(F\) be a field and let \(A=F\langle t_1,\dots,t_n\rangle\) be a free algebra with generators \(t_1,\dots,t_n\). Let us denote by \(T\) the commutator ideal of \(A\). Then \(C=A/T\) is a ring of polynomials and \(B=A/T^2\) is a free metabelian algebra. Let \(C'\) be the algebra which is anti-isomorphic to \(C\). The author introduces a ``gradient'' from \(B\) to \((C'\otimes_FC)^n\) and with the help of this gradient connects with every endomorphism of \(B\) a matrix with elements from \(C'\otimes_FC\) which is analogous to the Jacobi matrix. He shows that an endomorphism of \(B\) is an automorphism if and only if the corresponding matrix is invertible over \(C'\otimes_FC\). He also proves that any automorphism of the polynomial ring is induced by an automorphism of the corresponding free metabelian algebra.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16S10
zbMATH DE Number: 923547
zbMATH Keywords: free algebras
zbMATH Keywords: polynomial rings
zbMATH Keywords: free metabelian algebras
zbMATH Keywords: endomorphisms
zbMATH Keywords: Jacobi matrix
zbMATH Keywords: automorphisms
author: Ualbai Umirbaev
reviewed by: Leonid Makar-Limanov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02107336
OpenAlex ID: W2019606581
description: scientific article; zbMATH DE number 923547