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)\circ_n d(y)=x\circ _m y (ii)~d(x)\circ_m d(y)=(d(x\circ y))^n 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.

Keywords:

Prime and semiprime rings , Derivations , Martindale ring of quotients , Banach algebras , Radical