Generalized Skew Derivations and g-Lie Derivations of Prime Rings

Abstract

Let R be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and C its extended centroid. Suppose that F is a nonzero generalized skew derivation of R, with the associated automorphism $\alpha$, and $p(x_1,\ldots ,x_n)$ a noncentral polynomial over C, such that

$$F\biggl ([x,y]\biggr )=[F(x),\alpha (y)]+[\alpha (x),F(y)]$$

for all $x,y \in \{p(r_1,\ldots ,r_n) : r_1,\ldots ,r_n \in R\}$. Then $\alpha$ is the identity map on R and F is an ordinary derivation of R.