On b-generalized derivations and commutativity of prime rings

Let A be a prime ring, Z(A) its center, Q its right Martindale quotient ring, C its extended centroid, ψ a non-zero b-generalized derivation of A with associated map ξ. In this article, we prove that: (i) If [ψ(x), ψ(y)] = 0 for all x, y ∈ A, then A is either commutative or there exists q ∈ Q such that ξ = ad(q), ψ(x) = -bxq, and qb = 0. (ii) If ψ(x) ◦ ψ(y) = 0 for all x, y ∈ A, then A is either commutative with char(A) = 2 or there exists q ∈ Q such that ψ(x) = -bxq and qb = 0. Additional results are established for cases involving [ξ(x), ψ(x)] = 0 or ξ(x)◦ψ(x) = 0, where char(A) = 2. Furthermore, we give some examples that show the importance of the hypotheses of our theorems.