A note on derivations in rings and Banach algebras

Document Type : Research Paper


1 Department of mathematics, Aligarh Muslim University, Aligarh, India.

2 School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui Province, P.R.China

3 Department of Mathematics, Faculty of Science and Arts-Rabigh, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia.



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.


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