Stable finiteness of group rings in arbitrary characteristic
type: Item
instance of: scholarly article
title: Stable finiteness of group rings in arbitrary characteristic
zbMATH Open document ID: 1018.20006
author: Pere Ara
author: K. C. O'Meara
author: Francesc Perera
published in: Advances in Mathematics
publication date: 2002-12-16T00:00:00Z
review text: A ring \(R\) is said to be directly finite (resp. stably finite) if one-sided inverses in \(R\) (resp. in all matrix rings \(M_n(R)\)) are two sided. It has long been known that the group algebra \(K[G]\) of any group \(G\) over a field \(K\) of characteristic zero is directly finite [\textit{I. Kaplansky}, Fields and Rings, Chicago Lectures in Mathematics, Univ. of Chicago Press, Chicago (1970; Zbl 0238.16001); \textit{M. S. Montgomery}, Bull. Am. Math. Soc. 75, 539-540 (1969; Zbl 0174.31204); \textit{D. S. Passman}, Proc. Am. Math. Soc. 28, 371-374 (1971; Zbl 0217.07802)]. On the other hand the direct finiteness of \(K[G]\) in the case when the characteristic of \(K\) is \(p>0\) has been an open problem. In the present paper the authors prove that over a division ring \(D\) of any characteristic, and for any free-by-amenable group \(G\), the group ring \(D[G]\) is stably finite. This is achieved by showing that for each finitely generated subgroup \(S\) of such a group \(G\), the group ring \(D[S]\) admits a faithful Sylvester rank function which implies that \(D[G]\) is stably finite. The proof makes an interesting use of the translation rings associated with amenable groups.
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 1845951
zbMATH Keywords: group rings
zbMATH Keywords: Sylvester rank functions
zbMATH Keywords: translation rings
zbMATH Keywords: stable finiteness
zbMATH Keywords: direct finiteness
zbMATH Keywords: division rings
zbMATH Keywords: free-by-amenable groups
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
full work available at URL: aima.2002.2075
OpenAlex ID: W2022354771
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DOI: 10.1006/AIMA.2002.2075
description: scientific article; zbMATH DE number 1845951