Free Division Rings of Fractions of Crossed Products of Groups With Conradian Left-Orders

TL;DR

This paper characterizes when division rings of fractions of crossed product group rings with Conradian left-orders are free, showing they embed into endomorphism rings of formal power series and are unique up to isomorphism.

Contribution

It introduces a notion of freeness for division rings of fractions relative to Conradian orders and establishes conditions for their embedding and uniqueness.

Findings

Division rings of fractions are free iff they embed into endomorphism rings of formal power series.

All such free division rings are isomorphic regardless of the chosen Conradian order.

The existence of a free division ring of fractions is equivalent to the rational closure being a skew field.

Abstract

Let $D$ be a division ring of fractions of a crossed product $F[G,\eta,\alpha]$ where $F$ is a skew field and $G$ is a group with Conradian left-order $\leq$. For $D$ we introduce the notion of freeness with respect to $\leq$ and show that $D$ is free in this sense if and only if $D$ can canonically be embedded into the endomorphism ring of the right $F$-vector space $F((G))$ of all formal power series in $G$ over $F$ with respect to $\leq$. From this we obtain that all division rings of fractions of $F[G,\eta,\alpha]$ which are free with respect to at least one Conradian left-order of $G$ are isomorphic and that they are free with respect to any Conradian left-order of $G$. Moreover, $F[G,\eta,\alpha]$ possesses a division ring of fraction which is free in this sense if and only if the rational closure of $F[G,\eta,\alpha]$ in the endomorphism ring of the corresponding right $F$-vector space $F((G))$ is a skew field.