Φ-derivations and commutativity of rings and algebras

Abstract

The main purpose of this paper is to investigate the effect of Φ-derivatives on the 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 d Φi = 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,a_{0}) = e for some a_{0} ⋲ A, where e is the unity of A. If {a ⋲ A : ψ(a, a_0) = 0} = {0}, then A is commutative. Some other related results are also discussed