derivations in semiprime rings and banach algebras

Abstract. 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 fixed 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 ∈ r where 0 ̸= b ∈ 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.