Universal localizations embedded in power-series rings

Abstract

Let R be a ring, let F be a free group, and let X be a basis of F. Let †: RF ! R denote the usual augmentation map for the group ring RF, let X@ := fx i 1 j x 2 Xg µ RF, let § denote the set of matrices over RF that are sent to invertible matrices by †, and let (RF)§ i1 denote the universal localization of RF at §. A classic result of Magnus and Fox gives an embedding of RF in the power-series ring RhhX@ii. We show that if R is a commutative Bezout domain, then the division closure of the image of RF in RhhX@ii is a universal localization of RF at §. We also show that if R is a von Neumann regular ring or a commutative Bezout domain, then (RF)§ i1 is stably flat as an RF-ring, in the sense of