Completed group algebras without zero divisors
type: Item
instance of: scholarly article
title: Completed group algebras without zero divisors
zbMATH Open document ID: 0669.16007
DOI: 10.1007/BF01261969
author: Andreas Neumann
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let p be a prime, and let G be a torsion free analytic pro-p group; that is, G is a torsion free pro-p group admitting the structure of an analytic manifold over \(Q_ p\) with analytic group operations. By a result of \textit{A. Lubotzky} and \textit{A. Mann} [J. Algebra 105, 484-505, 506-515 (1987; Zbl 0626.20010, Zbl 0626.20022)], a pro-p group is analytic if and only if there is a finite bound on the cardinality of a generating set of its open subgroups. (Such groups include the poly-p- adic groups, those pro-p groups having a finite subnormal series of closed subgroups \(1=G_ 0\subset...G_ i\subset...\subset G_ n=G\) with \(G_{i+1}/G_ i\) isomorphic either to the p-adic integers \({\mathbb{Z}}_ p\) or to the cyclic group of order p, for all p.) Let \({\mathbb{Z}}_ pG\) denote the completion \(\lim_{\leftarrow}{\mathbb{Z}}_ p[G/N]\), where G/N ranges over the finite images of G. The main result of this note is that \({\mathbb{Z}}_ pG\) is a domain. Since \({\mathbb{Z}}_ p[G]\) embeds in \({\mathbb{Z}}_ pG\), the same conclusion applies to the ordinary group ring too. Being a projective limit of local rings, \(Z_ pG\) is a local ring. Appealing largely to results of \textit{M. Lazard} [Publ. Math., Inst. Hautes Etud. Sci. 26, 389-603 (1965; Zbl 0139.023)], the author shows that \({\mathbb{Z}}_ pG\) is Noetherian of finite global dimension. The desired conclusion now follows from a result of \textit{R. Walker} [Proc. Lond. Math. Soc., III. Ser. 24, 27-45 (1972; Zbl 0224.16004)].
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 4095672
zbMATH Keywords: zero divisors
zbMATH Keywords: torsion free analytic pro-p group
zbMATH Keywords: open subgroups
zbMATH Keywords: finite subnormal series of closed subgroups
zbMATH Keywords: completion
zbMATH Keywords: finite images
zbMATH Keywords: domain
zbMATH Keywords: group ring
zbMATH Keywords: projective limit of local rings
zbMATH Keywords: finite global dimension
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Noetherian Rings of Finite Global Dimension
cites work: A Change of Rings Theorem and the Artin-Rees Property
cites work: \(K_0\) and noetherian group rings
cites work: Q5515432
cites work: Projective modules
cites work: \(p\)-adic analytic groups
cites work: Powerful \(p\)-groups. I: Finite groups.
cites work: Sur la dimension cohomologique des groupes profinis
cites work: Local Rings and Normalizing Sets of Elements
cites work: Q4145843
description: scientific article; zbMATH DE number 4095672