Journal of the
Korean Mathematical Society JKMS

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $n$ a fixed positive integer. (i) If $(d(x)y+xd(y)+d(y)x+yd(x))^{n}=xy+yx$ for all $x,y\in I$, then $R$ is commutative. (ii) If ${\rm char}R\neq 2$ and $(d(x)y+xd(y)+d(y)x+yd(x))^{n}-(xy+yx)$ is central for all $x,y\in I$, then $R$ is commutative. We also examine the case where $R$ is a semiprime ring.

Keywords: prime and semiprime rings, left Utumi quotient rings, differential identities, derivations

MSC numbers: Primary 16N60; Secondary 16W25