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Universal localizations embedded in power-series rings
-
Pere Ara
and
Warren Dicks
Published/Copyright:
March 12, 2007
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.
Received: 2006-01-29
Accepted: 2006-03-30
Published Online: 2007-03-12
Published in Print: 2007-03-20
© Walter de Gruyter
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Articles in the same Issue
- P-adic logarithmic forms and group varieties III
- Moving homology classes to infinity
- On a class of locally finite T-groups
- On Ext-universal modules in Gödel's universe
- Paraproducts in one and several parameters
- Labeled configuration spaces and group completions
- Universal localizations embedded in power-series rings
Articles in the same Issue
- P-adic logarithmic forms and group varieties III
- Moving homology classes to infinity
- On a class of locally finite T-groups
- On Ext-universal modules in Gödel's universe
- Paraproducts in one and several parameters
- Labeled configuration spaces and group completions
- Universal localizations embedded in power-series rings
