Characterizations of Lie derivations of triangular algebras

AbstractLet T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if δ:T→T is an R-linear map satisfyingδ([x,y])=[δ(x),y]+[x,δ(y)]for any x,y∈T with xy=0 (resp. xy=p, where p is the standard idempotent of T), then δ=d+τ, where d is a derivation of T and τ:T→Z(T) (where Z(T) is the center of T) is an R-linear map vanishing at commutators [x,y] with xy=0 (resp. xy=p)