A note on derivations in rings and Banach algebras

Let R be a prime ring with U the Utumi quotient ring and Q be the Martindale quotient ring of R, respectively. Let d be a derivation of R and m,n be fixed positive integers. In this paper, we study the case when one of the following holds: (i) d(x) ◦n d(y)=x ◦m y (ii) d(x) ◦m d(y)=(d(x ◦ y)) for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on Banach algebras.