Characterizations of Lie n-derivations of unital algebras with nontrivial idempotents

Abstract

Let A be a unital algebra with a nontrivial idempotent e, and f=1-e. Suppose that A satisfies that exe.eAf={0}=fAe.exe implies exe=0 and eAf.fxf={0}=fxf.fAe implies fxf=0 for each x in A. For a Lie n-derivation φ on A, we obtain the necessary and sufficient conditions for φ to be standard, i.e., φ=d+γ, where d is a derivation on A, and γ is a linear mapping from A into the centre Z(A) vanishing on all (n-1)-th commutators of A. Furthermore, we also consider the sucient conditions under which each Lie n-derivation on A can be standard.