# Universal localizations embedded in power-series rings

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5 Citations (Scopus)

## Abstract

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. © Walter de Gruyter 2007.
Original language English 365-378 Forum Mathematicum 19 https://doi.org/10.1515/FORUM.2007.015 Published - 20 Mar 2007