A note on division rings satisfying generalized rational identities with anti-automorphisms

TL;DR

We study division rings $D$ with infinite center $F$ satisfying a generalized rational identity with respect to an anti-automorphism $\\sigma$ (order constraint $\\sigma^m \neq \\mathrm{Id}$). Our approach develops notation for $D\\langle X_m\\rangle$, $D(X_m)$, and blended $\\sigma^m$-GPIs, proves a key blended $\\sigma^m$-linear GPI lemma, and applies the PI/GPI/GRI equivalence to deduce central finiteness. This yields an anti-automorphism counterpart of Amitsur's theorem: if $D$ satisfies a $\\sigma^m$-GRI, then $D$ is centrally finite. The results extend prior work on involutions and generalized identities, providing criteria for finite-dimensionality over the center in the presence of nontrivial anti-automorphisms.

Abstract

Let $D$ be a division ring with infinite center $F$; $σ$ be an anti-automorphism of $D$ and $m$ be a positive integer such that $σ^m\neq \mathrm{Id}$. In this paper, we show that if $D$ satisfies a $σ^m$-GRI, then $D$ is centrally finite.

Theorems & Definitions (14)

• theorem 1
• proposition 1
• lemma 1
• proof
• corollary 1
• lemma 2
• proof
• lemma 3
• proof
• proposition 2