Free algebras and free groups in Ore extensions and free group algebras in division rings

TL;DR

This paper investigates conditions under which division rings constructed as Ore extensions contain free algebras, showing that either they satisfy polynomial identities or contain free subalgebras, with applications to division rings with certain solvable subgroups.

Contribution

It establishes a dichotomy for division rings formed via Ore extensions, linking free subalgebra existence to properties of automorphisms and derivations, and applies this to broader classes of division rings.

Findings

Division rings from Ore extensions either satisfy polynomial identities or contain free algebras.

Existence of free subalgebras in such rings relates to automorphism order and derivation non-triviality.

Division rings with certain solvable subgroups contain free algebras or free group algebras.

Abstract

Let $K$ be a field of characteristic zero, let $\sigma$ be an automorphism of $K$ and let $\delta$ be a $\sigma$-derivation of $K$. We show that the division ring $D=K(x;\sigma,\delta)$ either has the property that every finitely generated subring satisfies a polynomial identity or $D$ contains a free algebra on two generators over its center. In the case when $K$ is finitely generated over $k$ we then see that for $\sigma$ a $k$-algebra automorphism of $K$ and $\delta$ a $k$-linear derivation of $K$, $K(x;\sigma)$ having a free subalgebra on two generators is equivalent to $\sigma$ having infinite order, and $K(x;\delta)$ having a free subalgebra is equivalent to $\delta$ being nonzero. As an application, we show that if $D$ is a division ring with center $k$ of characteristic zero and $D^*$ contains a solvable subgroup that is not locally abelian-by-finite, then $D$ contains a free $k$-algebra on two generators. Moreover, if we assume that $k$ is uncountable, without any restrictions on the characteristic of $k$, then $D$ contains the $k$-group algebra of the free group of rank two.