Characterization of Lie-type higher derivations of triangular rings

Let A be a triangular ring and let pn (U1, U2, Un) denote the (n - 1) th commutator of elements U1, U2, Un ∈ A. Suppose that N is the set of nonnegative integers and L = { ζ r } r ∈ N is a sequence of additive mappings on A such that ζ 0 = id A, the identity mapping on A, and for each r ∈ N, ζ r (pn (U1, U 2, Un)) = ς i 1 + i 2 + i n = r pn (ζ i1 (U 1), ζ i2 (U2), ζ in (Un)) for all U1, U 2, Un⋯ with U 1 U2 · Un = 0. In this paper, it is shown that under certain conditions L = { ζ r } r ∈ N has the standard form, that is, there exist a higher derivation {dr} r ∈ N on A and a family {hr } r ∈ N of additive mappings h r: A → Z (A) satisfying hr(pn (U1, U2, Un)) = 0 for all U1, U2, U nA with U1 U2 ⋯ Un = 0 such that for each r ∈ N, ζ r (U) = dr (U) + hr (U) for all U A.