On generalized higher derivations on semiprime rings

Abstract

We obtain the following result: Let R be a 2-torsion free semiprime ∗-ring. Suppose there exists a family of additive mappings F=(f_i)_i∈ℕ0 of ℝ associated with some higher derivation D = (d_i)_i∈ℕ0 of R, such that f₀ = id_R and the relation f_n(xx^∗) = ∑_i+j=nf_i(x)d_j(x^∗) is fulfilled for each n ∈ ℕ₀ and for all x ∈ R. Then F is a generalized higher derivation. Further, we also prove that if the relation f_n(xy^∗x) = ∑_i+j+k=n f_i(x)d_j(y^∗)d_k(x) is fulfilled for each n ∈ ℕ₀ and for all x,y ∈ R, then F is also a generalized higher derivation.