φ-derivations and commutativity of rings and algebras

The main purpose of this paper is to investigate the effect of φ-derivatives on commutativity of rings and algebras. let r be a 2-torsion free prime ring, d : r → r be a φ-derivation such that φ is an epimorphism and dφ = φd = d. if [φ(a), φ(x)]d(y) = d(x)[y, a] for all x, y, a ∈ r, then r is commutative or d is zero. another result in this regard reads as follows: let (a, ∗) be a unital, involutive algebra, and let ψ : a × a → a be a ∗-two variable φ-derivation such that ψ(e, a0) = e for some a0 ∈ a, where e is the unity of a. if {a ∈ a : ψ(a, a0) = 0} = {0}, then a is commutative.