Communications of the
Korean Mathematical Society CKMS

Let $R$ be a $2$-torsion free prime ring and $J$ be a nonzero Jordan ideal of $R$. Let $F$ and $G$ be two generalized derivations with associated derivations $f$ and $g$, respectively. Our main result in this paper shows that if $F(x)x-xG(x)=0$ for all $x\in J$, then $R$ is commutative and $F=G$ or $ G$ is a left multiplier and $F=G+f$. This result with its consequences generalize some recent results due to El-Soufi and Aboubakr in which they assumed that the Jordan ideal $J$ is also a subring of $R$.

Keywords: prime rings, generalized derivations, Jordan ideals

MSC numbers: 16W10, 16W25, 16U80