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 (xm)xn + xnF (xm),xr]k = 0 for all x E I, where m, n, r, k are fixed positive integers, then there exists & lambda; E C such that F (x) = & lambda;x unless R & SIM;= M2(GF(2)), the 2 x 2 matrix ring over the Galois field GF(2) of two elements.