MMN-1217
On ideals with skew derivations of prime rings
Abstract
Let $R$ be a prime ring and set $[x,y]_1=[x,y]=xy-yx$ for all $x,y\in R$ and inductively $[x,y]_k=[[x,y]_{k-1},y]$ for $k>1$. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let $R$ be a prime ring and $I$ a nonzero ideal of $R$. Suppose that $(\delta,\varphi)$ is a skew derivation of $R$ such that $\delta([x,y])=[x,y]_n$ for all $x,y \in I$, then $R$ is commutative.
Vol. 15 (2014), No. 2, pp. 717-724