Valuation methods in group rings and in skew fields. I
type: Item
instance of: scholarly article
title: Valuation methods in group rings and in skew fields. I
zbMATH Open document ID: 1022.16034
DOI: 10.1016/S0021-8693(02)00042-X
author: A. I. Lichtman
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The author is the expert on applying valuation theoretic methods to group rings, enveloping algebras of Lie algebras, and skew fields. In this paper, he constructs a family of discrete valuations in group rings of residually torsion-free nilpotent groups and extends these valuations to the Malcev-Neumann power series skew field of these group rings. The main results here are quite technical in nature, so we mention just two of the many theorems in this long paper. Theorem~I. Let \(H\) be a group with an N-series \(\{H_i\}\) and let \(K\) be a (commutative) field of characteristic \(p\). Assume that \(\bigcap_{i=1}^\infty H_i=1\) and that all factors \(H_i/H_{i+1}\) are torsion-free Abelian with no elements of infinite \(p\)-height. Then, using the filtration defined by the series, we have \(\text{gr}(KH)\cong K\otimes\text{gr}(ZH)\). This, in turn, yields results on the enveloping algebras of certain Lie algebras associated to \(H\). Theorem~III. Let \(R\) be a ring and let \((t)\) be the ideal generated by the central regular element \(t\), with \(\bigcap_{n=1}^\infty(t)^n=0\). If \(\overline R=R/(t)\) is a semiprime Goldie ring, then the \(t\)-completion \(\widetilde R\) of \(R\) can be embedded in a semiprime Artinian ring. Furthermore, if \(\overline R\) is an Ore domain with skew field of fractions \({\mathcal D}(\overline R)\), then \(R\) can be embedded in a skew field \({\mathcal D}(R)\) having a \(t\)-adic valuation with residue skew field \({\mathcal D}(\overline R)\). This is a key ingredient in the proofs of several of the main group ring results.
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 1868058
zbMATH Keywords: valuation methods
zbMATH Keywords: group rings
zbMATH Keywords: skew fields
zbMATH Keywords: residually torsion-free nilpotent groups
zbMATH Keywords: Malcev-Neumann power series
zbMATH Keywords: N-series
zbMATH Keywords: enveloping algebras
zbMATH Keywords: filtrations
zbMATH Keywords: completions
reviewed by: D. S. Passman
MaRDI profile type: Publication
cites work: Observations on group rings
cites work: Q3218301
cites work: Sur les groupes nilpotents et les anneaux de Lie
cites work: Q3503673
cites work: On Embedding of Group Rings of Residually Torsion Free Nilpotent Groups into Skew Fields
cites work: Valuation methods in division rings
cites work: The residual nilpotence of the multiplicative group of a skew field generated by universal enveloping algebras
cites work: Free subgroups in linear groups over some skew fields
cites work: On the Embedding of Rings in Skew Fields
cites work: On universal fields of fractions for free algebras
cites work: On the associated graded ring of a group ring
cites work: Q4494221
cites work: Matrix rings and linear groups over a field of fractions of enveloping algebras and group rings. I
cites work: Q3786586
cites work: Localization and the AR Property
cites work: Q5813687
cites work: Q3882615
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description: scientific article; zbMATH DE number 1868058