Characterizations of Lie derivations of triangular algebras

Let T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if delta : T -> T is an R-linear map satisfying delta([x, y]) = [delta(x), y] + [x, delta(y)] for any x, y is an element of T with xy = 0 (resp. xy = p, where p is the standard idempotent of T), then delta = d + tau, where d is a derivation of T and tau : 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). Lie derivation (C) 2011 Elsevier Inc. All rights reserved.