Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$, $m (\geq 1), n (\geq 1)$ two fixed integers and $a\in R$. (i) If $a((d(x)y+xd(y)+d(y)x+yd(x))^{n}-(xy+yx))^{m}=0$ for all $x,y\in I$, then either $a=0$ or $R$ is commutative; (ii) If char$(R)\neq 2$ and $a((d(x)y+xd(y)+d(y)x+yd(x))^{n}-(xy+yx))\in Z(R)$ for all $x,y\in I$, then either $a=0$ or $R$ is commutative.

Keywords: prime ring, derivation, extended centroid

MSC numbers: 16W25, 16R50, 16N60