Characterizations of Lie derivations of triangular algebras

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