Abstract

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 d₀ = I_R, the identity map on R and 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, f₀ = I_R and (resp. 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.

Keywords:

Derivation; Higher derivation; Jordan - higher derivation; Lie ideal