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.