Yazd University Algebraic Structures and Their Applications 2382-9761 6 1 2019 03 01 A note on derivations in rings and Banach algebras 115 125 1378 10.22034/as.2019.1378 EN Nadeem Ur Rehman Department of mathematics, Aligarh Muslim University, Aligarh, India. 0000-0003-3955-7941 Shuliang Huang School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui Province, P.R.China Mohd Arif Raza Department of Mathematics, Faculty of Science and Arts-Rabigh, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia. 0000-0001-6799-8969 Journal Article 2019 02 28 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:<br />$(i)$~ $d(x)\circ_n d(y)$=$x\circ _m y$ $(ii)$~$d(x)\circ_m d(y)$=$(d(x\circ y))^n$ 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. https://as.yazd.ac.ir/article_1378_661a490ef5fc9de579717cacdf1c76ab.pdf