ONE-SIDED GENERALIZED (α, β)−REVERSE DERIVATIONS OF ASSOCIATIVE RINGS

In this paper, we introduce the notion of the one-sided generalized (α, β)−reversederivation of a ring R. Let R be a semiprime ring, ϱ be a non-zero ideal of R, α bean epimorphism of ϱ, β be a homomorphism of ϱ (α be a homomorphism of ϱ, βbe an epimorphism of ϱ) and γ : ϱ → R be a non-zero (α, β)−reverse derivation.We show that there exists F : ϱ → R, an l−generalized (α, β)−reverse derivation(an r−generalized (α, β)−reverse derivation) associated with γ iff F(ϱ), γ(ϱ) ⊂ CR(ϱ)and F is an r−generalized (β, α)−derivation (an l−generalized (β, α)−derivation) associated with (β, α)−derivation γ on ϱ. This theorem generalized the results of A.Aboubakr and S. Gonzalez proved in [1, Theorem 3.1 and Theorem 3.2 ].