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On generalized Lie derivations. (English) Zbl 1449.16077

Summary: In this paper, we investigate generalized Lie derivations. We give a complete characterization of when each generalized Lie derivation is a sum of a generalized inner derivation and a Lie derivation. This generalizes a result given by D. Benkovič [Linear Algebra Appl. 434, No. 6, 1532–1544 (2011; Zbl 1216.16032)]. We also investigate when every generalized Lie derivation on some particular kind of unital algebras is a sum of a generalized derivation and a central map which vanishes on all commutators. Precisely, we consider both the unital algebras with nontrivial idempotents and the trivial extension algebras.

MSC:

16W25 Derivations, actions of Lie algebras
47B47 Commutators, derivations, elementary operators, etc.
15A78 Other algebras built from modules

Citations:

Zbl 1216.16032
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References:

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