Bulletin of the
Korean Mathematical Society BKMS

In this paper we will show that if there exist derivations $D$, $G$ on a $n!$-torsion free semi-prime ring $R$ such that the mapping $D^{2}+G$ is n-commuting on $R$, then $D$ and $G$ are both commuting on $R$. And we shall give the algebraic conditions on a ring that a Jordan derivation is zero.

Keywords: torsion free ring, derivation, commuting, $n$-commuting, Jordan derivation, prime, semi-prime, semisimple, noncommutative Banach algebras

MSC numbers: 16A12, 16A70, 16A72, 46H40, 46J10, 47B47