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

الأصل · EN

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 ϕ on A to be of the form ϕ=d+δ+γ, where d is a derivation on A, δ is a singular Jordan 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. In particular, we also discuss the (necessary and) sufficient conditions for a Lie n-derivation ϕ on A to be standard, i.e., ϕ=d+γ.