Ore extensions which are GPI-rings

Abstract

Let R be a prime ring and δ a σ-derivation of R, where σ is an automorphism of R. It is proved that the skew polynomial ring $R[t; \sigma, \delta]$ is a GPI-ring (PI-ring resp.) if and only if R is a GPI-ring (PI-ring resp.), δ is quasi-algebraic, and σ is quasi-inner. If $R[t; \sigma, \delta]$ is a GPI-ring then soc $(Q[t;\sigma,\delta]\tilde C) = \Big({\rm soc}(Q)[t;\sigma,\delta]\Big)\tilde C$ , where Q is the symmetric Martindale quotient ring of R and where $\tilde C$ denotes the extended centroid of $Q[t; \sigma, \delta]$ . If $R[t; \sigma, \delta]$ is a PI-ring, its PI-degree is determined as follows: $(1) {\rm PI-deg}(R[t; \sigma, \delta]) = {\rm PI-deg}(R) \times {\rm out-deg}(\delta)$ if δ is X-outer, and $(2) {\rm PI-deg}(R[t; \sigma, \delta]) = {\rm PI-deg}(R) \times {\rm out-deg}(\sigma)$ if δ is X-inner.