On the Commutativity of a Prime ∗-Ring with a ∗-α-Derivation

Abstract

Let R be a prime ∗-ring where ∗ be an involution of R, α be an automorphism of R, T be a nonzero left α-∗-centralizer on R and d be a nonzero ∗-α-derivation on R. The aim of this paper is to prove the commutativity of a ∗-ring R with the followings conditions: i) if T is a homomorphism (or an antihomomorphism) on R,ii) if d([x, y]) = 0 for all x, y ∈ R, iii) if [d(x), y] = [α(x), y] for all x, y ∈ R, iv) if d(x) ◦ y = 0 for all x, y ∈ R, v) if d(x ◦ y) = 0 for all x, y ∈ R.