Characterizations of Lie Triple Higher Derivations of Triangular Algebras by Local Actions
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Let N be the set of nonnegative integers and A be a 2-torsion free triangular algebra over a commutative ring R. In the present paper, under some lenient assumptions on A, it is proved that if ∆ = {δn}n∈N is a sequence of R-linear mappings δn : A → A satisfying δn([[x, y], z]) = Pi+j+k=n [[δi(x), δj (y)], δk(z)] for all x, y, z ∈ A with xy = 0 (resp. xy = p, where p is a nontrivial idempotent of A), then for each n ∈ N, δn = dn +τn; where dn : A → A is R-linear mapping satisfying dn(xy) = P i+j=n di(x)dj (y) for all x, y ∈ A, i.e. D = {dn}n∈N is a higher derivation on A and τn : A → Z(A) (where Z(A) is the center of A) is an R-linear map vanishing at every second commutator [[x, y], z] with xy = 0 (resp.xy = p).
Abstract
Let N be the set of nonnegative integers and A be a 2-torsion free triangular algebra over a commutative ring R. In the present paper, under some lenient assumptions on A, it is proved that if ∆ = {δn}n∈N is a sequence of R-linear mappings δn : A → A satisfying δn([[x, y], z]) = Pi+j+k=n [[δi(x), δj (y)], δk(z)] for all x, y, z ∈ A with xy = 0 (resp. xy = p, where p is a nontrivial idempotent of A), then for each n ∈ N, δn = dn +τn; where dn : A → A is R-linear mapping satisfying dn(xy) = P i+j=n di(x)dj (y) for all x, y ∈ A, i.e. D = {dn}n∈N is a higher derivation on A and τn : A → Z(A) (where Z(A) is the center of A) is an R-linear map vanishing at every second commutator [[x, y], z] with xy = 0 (resp.xy = p).
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