Derivations in semiprime rings and Banach algebras

Let R be a 2-torsion free semiprime ring with extended centroid C, U the Utumi quotient ring of R and m; n > 0 are xed integers. We show that if R admits derivation d such that b[d(x); x]n; [y; d(y)]m] = 0 for all x; y 2 R where 0 ̸= b 2 R, then there exists a central idempotent element e of U such that eU is commutative ring and d induce a zero derivation on (1 e)U. We also obtain some related result in case R is a non-commutative Banach algebra and d continuous or spectrally bounded.