A note on locally nilpotent derivations
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.
scientific article; zbMATH DE number 89561