An Engel condition with skew derivations for one-sided ideals

Abstract

We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and $\delta$ a nonzero $\sigma$-derivation of A, where $\sigma$ is an epimorphism of A. For $x,y\in A$, we set $[x,y] = xy - yx$. If $[[\ldots [[\delta (x^{n_0}),x^{n_1}],x^{n_{2}}],\ldots ],x^{n_k}]=0$ for all $x\in R$, where $n_{0},n_{1},\ldots ,n_{k}$ are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) $C\cong GF(2)$, the Galois field of two elements; (3) there exist $b\in Q$ and $\lambda \in C$ such that $\delta (x)=\sigma (x)b-bx$ for all $x\in A$, $(b-\lambda )R=0$ and $\sigma (R)=0$. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).