Nil derivations
Let F be a field of characteristic zero, R be a prime F-algebra and d be a derivation of R. Then d is said to be nilpotent if \(d^ n=0\) for some n and to be nil if for all \(x\in R\) there exists \(n=n(x)\) such that \(d^ n(x)=0\). If I is a non-zero ideal of R then R imbeds in the endomorphism ring E(I) of the right R-module I. If there exists a non-zero ideal and an element f of E(I) such that for all \(x\in R\) and \(w\in I\), \(d(x)(w)=(fx-xf)(w)\), d is said to be E-inner induced by f. The main theorem of the paper establishes criteria for a nil derivation to be E- inner induced by an element f which is 2-sided nil i.e. for each \(x\in R\) there exists \(n=n(x)\) such that \(xf^ n(y)=0=f^ n(x)\) for all \(y\in R\). It should be noted that in the polynomial algebra F[x], d/dx is nil but not E-inner and that there exist, in enveloping algebras of Lie algebras, examples of nil E-inner derivations which are not induced by nil elements. For nilpotent derivations the situation is simpler and, as a corollary of the main theorem, it is shown that a nilpotent derivation of R is E-inner induced by a nilpotent element. Related results of \textit{V. K. Kharchenko} [Algebra Logika 17, 220-238 (1978; Zbl 0423.16011)], \textit{Chung}, \textit{Kovacs} and \textit{Luh} [preprint] and \textit{W. Martindale} and \textit{C. R. Miers} [Pac. J. Math. 104, 179-190 (1983; Zbl 0507.16022)] are discussed as are some as yet unpublished results of the author.
scientific article; zbMATH DE number 3961890
scientific article