Finiteness and homological conditions in commutative group rings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1245.13007
author: Sarah Glaz
author: Ryan Schwarz
publication date: 2012-07-02T00:00:00Z
Mathematics Subject Classification ID: 13B99
Mathematics Subject Classification ID: 13D05
zbMATH DE Number: 6052219
zbMATH Keywords: group rings
zbMATH Keywords: Noetherian rings
zbMATH Keywords: coherent rings
zbMATH Keywords: finite conductor rings
zbMATH Keywords: weak global dimension
zbMATH Keywords: von Neumann regular rings
zbMATH Keywords: semihereditary rings
zbMATH Keywords: Prüfer conditions
zbMATH Keywords: zero divisors
zbMATH Keywords: PP tings
zbMATH Keywords: PF rings
MaRDI profile type: Publication
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title: Finiteness and homological conditions in commutative group rings
review text: This paper surveys finiteness properties and homological properties for commutative group rings. In the following, \(R\) denotes a commutative ring, \(G\) is an abelian group, and \(RG\) is the associated group ring.NEWLINENEWLINEIn Section 2, a number of finiteness properties for \(RG\) are characterized via finiteness properties for \(R\) and \(G\). For example, \(RG\) is noetherian (respectively, artinian) if and only if \(R\) is noetherian (respectively, artinian) and \(G\) is finitely generated (respectively, finite). Properties such as coherence and UFD are also discussed.NEWLINENEWLINEIn Section 3, some homological properties for \(RG\) are characterized via properties for \(R\) and \(G\). For example \(RG\) is von Neumann regular if and only if \(R\) is von Neumann regular, \(G\) is torsion, and \(R\) is uniquely divisible by the order of every element of \(G\). Properties such as finiteness of the weak global dimension and regularity are also discussed.NEWLINENEWLINESection 4 focuses on certain zero divisor controlling conditions. For example, \(RG\) is an integral domain if and only if \(R\) is an integral domain and \(G\) is torsion free. And if \(G\) is torsion free, then \(RG\) is a PF (respectively PP) ring if and only if \(R\) is a PF (respectively PP) ring.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13006].
reviewed by: Henrik Holm
description: scientific article; zbMATH DE number 6052219