A classification and examples of rank one chain domains

A chain order of a skew field $D$ is a subring $R$ of $D$ so that $d\in D\backslash R$ implies $d^{-1}\in R.$ Such a ring $R$ has rank one if $J(R)$, the Jacobson radical of $R,$ is its only nonzero completely prime ideal. We show that a rank one chain order of $D$ is either invariant, in which case $R$ corresponds to a real-valued valuation of $D,$ or $R$ is nearly simple, in which case $R,$ $J(R)$ and $(0)$ are the only ideals of $R,$ or $R$ is exceptional in which case $R$ contains a prime ideal $Q$that is not completely prime. We use the group $\mathcal{M}(R)$ of divisorial $R{\text{-}ideals}$ of $D$ with the subgroup $\mathcal{H}(R)$ of principal $R{\text{-}ideals}$ to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index $k$of $\mathcal{H}(R)$ in $\mathcal{M}(R).$Using the covering group $\mathbb{G}$ of $\operatorname{SL}(2,\mathbb{R} )$ and the result that the group ring $T\mathbb{G}$ is embeddable into a skew field for $T$ a skew field, examples of rank one chain orders are constructed for each possible exceptional case.