Which integral group rings are Bass orders?
type: Item
instance of: scholarly article
title: Which integral group rings are Bass orders?
zbMATH Open document ID: 0691.16008
DOI: 10.1016/0021-8693(90)90226-E
author: E. Kleinert
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: The group rings \({\hat {\mathbb{Z}}}_ pG\) over the complete ring \({\hat {\mathbb{Z}}}_ p\) of p-adic integers which are Bass orders are characterized as those for which G is a semidirect product of a \(p'\)-group by a p-group of order \(\leq p\). The proof is carried out by means of the classification of local Bass orders (Drozd, Kirichenko, Rojter) and \textit{S. A. Amitsur}'s work on finite subgroups of skewfields [Trans. Am. Math. Soc. 80, 361-386 (1955; Zbl 0065.256)]; it also implies that \({\hat {\mathbb{Z}}}_ pG\) is Bass if and only if its principal block is a Bass order. As a corollary, it follows that \({\mathbb{Z}}G\) is Bass if and only if G is cyclic of squarefree order.
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 4130578
zbMATH Keywords: group rings
zbMATH Keywords: p-adic integers
zbMATH Keywords: Bass orders
zbMATH Keywords: semidirect product
zbMATH Keywords: p-group
zbMATH Keywords: local Bass orders
zbMATH Keywords: finite subgroups of skewfields
zbMATH Keywords: principal block
MaRDI profile type: Publication
cites work: Finite Subgroups of Division Rings
cites work: Q5650865
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cites work: ON HEREDITARY AND BASS ORDERS
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cites work: The structure over \(\mathbb{Z}[q,q^{-1}]\) of Hecke orders of dihedral groups
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description: scientific article; zbMATH DE number 4130578