Nadeem Ur Rehman
Articles written in Proceedings – Mathematical Sciences
Volume 124 Issue 4 November 2014 pp 497-500
An Engel condition with an additive mapping in semiprime rings
Maja Fošner Nadeem Ur Rehman Joso Vukman
The main purpose of this paper is to prove the following result: Let $n \gt 1$ be a fixed integer, let 𝑅 be a $n!$-torsion free semiprime ring, and let $f : R \to R$ be an additive mapping satisfying the relation $[f (x), x]_{n} = [[... [[f(x),x],x],...], x] = 0$ for all $x \in R$. In this case $[f(x), x] = 0$ is fulfilled for all $x \in R$. Since any semisimple Banach algebra (for example, $C^{\ast}$ algebra) is semiprime, this purely algebraic result might be of some interest from functional analysis point of view.
Volume 126 Issue 3 August 2016 pp 389-398 Research Article
On prime and semiprime rings with generalized derivations and non-commutative Banach algebras
MOHD ARIF RAZA NADEEM UR REHMAN
Let $R$ be a prime ring of characteristic different from 2 and $m$ a fixed positive integer. If $R$ admits a generalized derivation associated with a nonzero deviation $d$ such that $[F(x), d(y)]_m = [x, y]$ for all $x$, $y$ in some appropriate subset of $R$, then $R$ is commutative. Moreover, we also examine the case $R$ is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer--Werner theorem.
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