Additive mappings covering generalized (alpha_1, alpha_2)-derivations in semiprime rings

Abstract

Let R be a semiprime ring, (α₁,α₂) be automorphisms on R and Δ, δ be additive maps from R to R. If Δ and δ satisfying any one of the following identities:
i) Δ(xⁿyⁿ)=Δ(xⁿ)α₁(yⁿ)+α₂(xⁿ)δ(yⁿ)
ii) Δ(xⁿ)=Δ(x^n-m)α₁(x^m)+α₂(x^n-m)δ(x^m), for all x,y∈R
then Δ will be generalized (α₁,α₂)-derivation with associated (α₁,α₂)-derivation on R.