B-generalized (α,β)-derivations and b-generalized (α,β)-biderivations of prime rings

Let R be a ring, α and β two automorphisms of R. An additive mapping d: R → R is called an (α, β) -derivation if d(xy) = d(x) α (y) + β (x) d(y) for any x,y ∈ R. An additive mapping G: R → R is called a generalized (α, β) -derivation if G(xy) = G(x) α (y) + β (x) d(y) for any x,y ∈ R, where d is an (α, β) -derivation of R. In this paper we introduce the definitions of b -generalized (α, β) -derivation and b -generalized (α, β) -biderivation. More precisely, let d: R → R and G: R → R be two additive mappings on R, α and β automorphisms of R and b ∈ R. G is called a b -generalized (α, β) -derivation of R, if G(xy) = G(x) α (y) + bβ (x) d(y) for any x,y ∈ R. Let now D: R × R → R be a biadditive mapping. The biadditive mapping Δ: R × R → R is said to be a b -generalized (α, β) -biderivation of R if, for every x,y,z ∈ R, Δ(x,yz) = Δ(x,y) α (z) + bβ (y) D(x,z) and Δ(xy,z) = Δ(x,z) α (y) + bβ (x) D(y,z). Here we describe the form of any b -generalized (α, β) -biderivation of a prime ring.