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 ε : RF → R denote the usual augmentation map for the group ring RF, let X∂ := {x − 1 | x ∈ X} ⊆ RF, let Σ denote the set of matrices over RF that are sent to invertible matrices by ε, and let (RF)Σ−1 denote the universal localization of RF at Σ. A classic result of Magnus and Fox gives an embedding of RF in the power-series ring R〈〈X∂〉〉. We show that if R is a commutative Bezout domain, then the division closure of the image of RF in R〈〈X∂〉〉 is a universal localization of RF at Σ. We also show that if R is a von Neumann regular ring or a commutative Bezout domain, then (RF)Σ−1 is stably flat as an RF-ring, in the sense of Neeman-Ranicki.