Rehman, N., Huang, S., Raza, M. (2019). A note on derivations in rings and Banach algebras. Algebraic Structures and Their Applications, 6(1), 115-125. doi: 10.29252/as.2019.1378

Nadeem ur Rehman; Shuliang Huang; Mohd Arif Raza. "A note on derivations in rings and Banach algebras". Algebraic Structures and Their Applications, 6, 1, 2019, 115-125. doi: 10.29252/as.2019.1378

Rehman, N., Huang, S., Raza, M. (2019). 'A note on derivations in rings and Banach algebras', Algebraic Structures and Their Applications, 6(1), pp. 115-125. doi: 10.29252/as.2019.1378

Rehman, N., Huang, S., Raza, M. A note on derivations in rings and Banach algebras. Algebraic Structures and Their Applications, 2019; 6(1): 115-125. doi: 10.29252/as.2019.1378

A note on derivations in rings and Banach algebras

^{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.

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.

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