Journal of the
Korean Mathematical Society JKMS

Let $R$ be a non-commutative prime ring and $I$ a non-zero left ideal of $R$. Let $U$ be the left Utumi quotient ring of $R$ and $C$ be the center of $U$ and $k,m,n,r$ fixed positive integers. If there exists a generalized derivation $g $ of $R$ such that $[g(x^{m})x^{n}, x^{r}]_{k}=0$ for all $x\in I$, then there exists $a\in U$ such that $g(x)=xa$ for all $x\in R$ except when $R \cong M_{2}(GF(2))$ and $I[I,I]=0$.

Keywords: prime rings, derivations, generalized derivations, left Utumi quotient rings, two-sided Martindale quotient ring, differential identities, Engel condition

MSC numbers: 16N60, 16W25, 16U80