Bull. Korean Math. Soc. 2002; 39(4): 635-643
Printed December 1, 2002
Copyright © The Korean Mathematical Society.
Ick-Soon Chang, Kil-Woung Jun, and Yong-Soo Jung
Chungnam National University, Chungnam National University, Chungnam National University
Our main goal is to show that if there exist Jordan derivations $D$ and $G$ on a noncommutative $(n+1)!$-torsion free prime ring $R$ such that $$ D(x)x^{n}-x^{n}G(x) \in C(R) $$ for all $x \in R$, then we have $D=0$ and $G=0$. We also prove that if there exists a derivation $D$ on a noncommutative 2-torsion free prime ring $R$ such that the mapping $x \mapsto [aD(x),x]$ is commuting on $R$, then we have either $a=0$ or $D=0$.
Keywords: noncommutative Banach algebra, derivation, commuting, prime ring, radical
MSC numbers: 47B47, 47B48, 16N20, 16N60
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