Invariants of Algebraic Automorphisms

Let R be a prime ring with extended centroid C and let σ be a C-algebraic automorphism of R. We let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R^{(\sigma)}\mathop{=}\limits^{\rm def.}\{x\in R\mid \sigma(x)=x\}$\end{document}, the subring of invariants of σ in R, and let Out-deg(σ) and Inn-deg(σ) denote the outer and inner degrees of σ, respectively. In the paper we first prove the nilpotence of the prime radical of R(σ) with a bound and characterize the semiprimeness and primeness of R(σ). Moreover, we show that if R(σ) is a prime PI-ring, then PI-deg(R) = PI-deg(R(σ)) × Inn-deg(σ).