On Generalized Jordan Triple (σ,τ)-Higher Derivations in Prime Rings

Abstract

Let R be a ring and let U be a Lie ideal of R. Suppose that σ,τ are endomorphisms of R, and ℕ is the set of all nonnegative integers. A family F={fn}n∈ℕ of mappings fn:R→R is said to be a generalized (σ,τ)-higher derivation (resp., generalized Jordan triple (σ,τ)-higher derivation) of R if there exists a (σ,τ)-higher derivation D={dn}n∈ℕ of R such that f0=IR, the identity map on R, fn(a+b)=fn(a)+fn(b), and fn(ab)=∑i+j=nfi(σn-i(a))dj(τn-j(b)) (resp., fn(aba)=∑i+j+k=nfi(σn-i(a))dj(σkτi(b))dk(τn-k(a))) hold for all a,b∈R and for every n∈ℕ. If the above conditions hold for all a,b∈U, then F is said to be a generalized (σ,τ)-higher derivation (resp., generalized Jordan triple (σ,τ)-higher derivation) of U into R. In the present paper it is shown that if U is a noncentral square closed Lie ideal of a prime ring R of characteristic different from two, then every generalized Jordan triple (σ,τ)-higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.