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. We obtain the (necessary and) sufficient conditions for a Lie n-derivation {\phi} on A to be of the form {\phi}=d+{\delta}+{\gamma}, where d is a derivation on A, {\delta} is a singular Jordan derivation on A and {\gamma} is a linear mapping from A into the centre Z(A) vanishing on all (n-1)-th commutators of A. In particular, we also discuss the (necessary and) sufficient conditions for a Lie n-derivation {\phi} on A to be standard, i.e., {\phi}=d+{\gamma}.