Alternative algebras admitting derivations with invertible values and invertible derivations. (English. Russian original) Zbl 1315.17025
Izv. Math. 78, No. 5, 922-936 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 5, 75-90 (2014).
In [J. Bergen et al., Can. J. Math. 35, 300–310 (1983; Zbl 0522.16031)] the structure of the associative rings admitting derivations with the property that all non-zero values are invertible was described.
In the paper under review an analogue of the Bergen-Herstein-Lanski theorem for alternative algebras is obtained.
Invertible derivations are also tried. The authors prove that a finite-dimensional alternative algebra over a field of characteristic zero is nilpotent if and only if it admits an invertible Leibniz derivation (the similar fact in the case of Lie algebras was proved in [W. A. Moens, Commun. Algebra 41, No. 7, 2427–2440 (2013; Zbl 1300.17015)]).
In the paper under review an analogue of the Bergen-Herstein-Lanski theorem for alternative algebras is obtained.
Invertible derivations are also tried. The authors prove that a finite-dimensional alternative algebra over a field of characteristic zero is nilpotent if and only if it admits an invertible Leibniz derivation (the similar fact in the case of Lie algebras was proved in [W. A. Moens, Commun. Algebra 41, No. 7, 2427–2440 (2013; Zbl 1300.17015)]).
Reviewer: Maria Trushina (Moskva)
MSC:
| 17D05 | Alternative rings |
| 17A36 | Automorphisms, derivations, other operators (nonassociative rings and algebras) |
