Universal localizations embedded in power-series rings

Let R be a ring, let F be a free group, and let X be a basis of F. Let epsilon : RF -> R denote the usual augmentation map for the group ring RF, let X partial derivative := {x - 1 vertical bar x epsilon X} subset of RF, let Sigma denote the set of matrices over RF that are sent to invertible matrices by epsilon, and let (RF)Sigma(-1) denote the universal localization of RF at Sigma. A classic result of Magnus and Fox gives an embedding of RF in the power-series ring R << X partial derivative >>. We show that if R is a commutative Bezout domain, then the division closure of the image of RF in R << X partial derivative >> is a universal localization of RF at Sigma. We also show that if R is a von Neumann regular ring or a commutative Bezout domain, then (RF)Sigma(-1) is stably flat as an RF-ring, in the sense of Neeman-Ranicki.