A note on locally nilpotent derivations
type: Item
instance of: scholarly article
title: A note on locally nilpotent derivations
zbMATH Open document ID: 0765.13004
DOI: 10.1016/0022-4049(92)90125-Y
author: Miguel Ferrero
author: Yves Lequain
author: Andrzej Nowicki
published in: Journal of Pure and Applied Algebra
publication date: 1993-01-16T00:00:00Z
review text: The authors call a derivation \(D\) of a ring \(R\) locally nilpotent if for each \(x\in R\) there is \(n(x)=n\geq 1\) with \(D^ n(x)=0\). The main result of this note proves that if \(R\) is a commutative \(\mathbb Z\)-torsion free ring with no nonzero nilpotent element, and if \(D\) and \(E\) are commuting, locally nilpotent derivations of \(R\) with \(D(b)=E(b)=0\) for some \(b\in R\), then \(aD+bE\) is a locally nilpotent derivation of \(R\) for \(a\in R\) exactly when \(D(a)=0\). The proof is an elementary but clever calculation done first when \(R\) is a domain. Examples show the necessity of the hypotheses, and the authors comment that the element \(b\in R\) in their result can be taken to be any unit. In corollary 5, the conditions that \(R\) is commutative and that \(D\) and \(E\) commute have been inadvertently omitted.
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13N99
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 89561
zbMATH Keywords: nil derivation
zbMATH Keywords: locally nilpotent derivation
MaRDI profile type: Publication
cites work: Q3255437
cites work: Automorphisms of polynomial and power series rings
cites work: Commutative bases of derivations in polynomial and power series rings
cites work: On the Jacobian conjecture
description: scientific article; zbMATH DE number 89561