A note on derivations in rings and Banach algebras
نویسندگان
1 School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui Province, P.R.China
2 Department of mathematics, Aligarh Muslim University, Aligarh, India.
3 Department of Mathematics, Faculty of Science and Arts-Rabigh, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia.
doi
10.22034/as.2019.1378چکیده
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.
