@article { author = {Rehman, Nadeem and Huang, Shuliang and Raza, Mohd}, title = {A note on derivations in rings and Banach algebras}, journal = {Algebraic Structures and Their Applications}, volume = {6}, number = {1}, pages = {115-125}, year = {2019}, publisher = {Yazd University}, issn = {2382-9761}, eissn = {2423-3447}, doi = {10.22034/as.2019.1378}, abstract = {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}, url = {https://as.yazd.ac.ir/article_1378.html}, eprint = {https://as.yazd.ac.ir/article_1378_661a490ef5fc9de579717cacdf1c76ab.pdf} }