An Engel condition with b-generalized derivations in prime rings

. Let R be a prime ring, I be a nonzero ideal of R , Q be its maximal right ring of quotients and C be its extended centroid. The aim of this paper is to show that if R admits a nonzero b -generalized derivation F such that [ F ( x m ) x n + x n F ( x m ) , x r ] k = 0 for all x ∈ I , where m , n , r , k are fixed positive integers, then there exists λ ∈ C such that F ( x ) = λ x unless R ∼ = M 2 ( GF ( 2 )) , the 2 × 2 matrix ring over the Galois field GF ( 2 ) of two elements.