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

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.