Group rings which are Jacobson rings
type: Item
instance of: scholarly article
title: Group rings which are Jacobson rings
zbMATH Open document ID: 0535.16013
DOI: 10.1007/BF01193776
author: Jan Krempa
author: Jan Okniński
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let A be a commutative ring and G an abelian group. In [\textit{G. Karpilovsky}, Commutative group algebras (1983; Zbl 0508.16010)] the following problem was stated (Problem 1): Find necessary and sufficient conditions for the group ring AG to be a Jacobson ring. In the case where the torsion-free rank of G is finite it is easy to prove that AG is a Jacobson ring if and only if A is. For a domain R let us denote by \(\hat R\) the field of fractions of R, and by fr(R) the smallest cardinality of a set of generators of \(\hat R\) as R-algebra. Our main result is Theorem: Let G be a group of infinite torsion-free rank \(\alpha\). Then AG is a Jacobson ring if and only if the following conditions are satisfied: 1) all homomorphic images of A have cardinality exceeding \(\alpha\), 2) for any prime homomorphic image B of A such that \(fr(B)\leq \alpha\) and for any ring C, \(B\subset C\subset \hat B,\) we have \(fr(C)\neq 1.\) The proof goes through a reduction to polynomial rings and is based on the fact that fr(R) is inherited by integral extensions of R.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 3849427
zbMATH Keywords: group ring
zbMATH Keywords: torsion-free rank
zbMATH Keywords: Jacobson ring
MaRDI profile type: Publication
cites work: Q5515450
cites work: The Nullstellensatz for Certain Group Rings
cites work: Hilbert rings and the Hilbert Nullstellensatz
cites work: Q5603302
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cites work: Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie
full work available at URL: https://doi.org/10.1007/bf01193776
OpenAlex ID: W1979482267
description: scientific article; zbMATH DE number 3849427