Commutative group algebras whose quotient rings by nilradicals are generated by idempotents.
type: Item
instance of: scholarly article
title: Commutative group algebras whose quotient rings by nilradicals are generated by idempotents.
zbMATH Open document ID: 1214.16018
DOI: 10.1216/RMJ-2011-41-1-229
author: Hideyasu Kawai
author: Nobuharu Onoda
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-03-01T00:00:00Z
review text: Let \(G\) be an Abelian group and \(R\) a commutative ring with identity. Denote by \(RG\) the group algebra of \(G\) over \(R\) and by \(\text{nil}(RG)\) the nil radical of \(RG\). The authors give necessary and sufficient conditions \(RG/\text{nil}(RG)\) to be generated by idempotents over \(R\) (Theorem 2.12). They prove (Theorem 2.18) that the following conditions are equivalent: (i) \(RG\) is generated by idempotents over \(R\) and (ii) \((R/\text{nil}(R))G\) is generated by idempotents over \(R/\text{nil}(R)\). The authors establish (Theorem 2.19) that if the ring \(R\) is indecomposable and \(\text{char}(R)=0\), then the condition (i) is equivalent to the following condition: (iii) \(RG/\text{nil}(RG)\) is generated by idempotents over \(R\).
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 5858887
zbMATH Keywords: commutative group algebras
zbMATH Keywords: nil radical
zbMATH Keywords: idempotents
zbMATH Keywords: indecomposable rings
zbMATH Keywords: Abelian groups
reviewed by: Todor Mollov
MaRDI profile type: Publication
cites work: Q5589390
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cites work: CONDITIONS FOR A PRODUCT OF RESIDUE-CLASS RINGS OF A RING TO BE GENERATED BY Ap-GROUP OF UNITS
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description: scientific article; zbMATH DE number 5858887