Generalized (σ , τ) higher derivations in prime rings

Let R be a ring and U be a Lie ideal of R. Suppose that σ, τ are endomorphisms of R. A family D = {d(n)}(n ∈ N)of additive mappings d(n):R → R is said to be a (σ,τ)- higher derivation of U into R if d(0) = I(R), the identity map on R and [Image: see text] holds for all a, b ∈ U and for each n ∈ N. A family F = {f(n)}(n ∈ N)of additive mappings f(n):R → R is said to be a generalized (σ,τ)- higher derivation (resp. generalized Jordan (σ,τ)-higher derivation) of U into R if there exists a (σ,τ)- higher derivation D = {d(n)}(n ∈ N)of U into R such that, f(0) = I(R) and [Image: see text] (resp. [Image: see text] holds for all a, b ∈ U and for each n ∈ N. It can be easily observed that every generalized (σ,τ)-higher derivation of U into R is a generalized Jordan (σ,τ)-higher derivation of U into R but not conversely. In the present paper we shall obtain the conditions under which every generalized Jordan (σ,τ)- higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.