Bulletin of the
Korean Mathematical Society BKMS

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