Carriers of torsion-free groups
type: Item
instance of: scholarly article
title: Carriers of torsion-free groups
zbMATH Open document ID: 0733.20033
author: R. S. Pierce
author: Charles Vinsonhaler
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1990-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108203
review text: The carrier of a torsion-free abelian group A is defined to be the set \(c(A)=\{p\in \Pi |\) pA\(\neq A\}\) where \(\Pi\) denotes the set of all rational primes. The number-theoretical properties of the primes in c(A) had played a rôle in the authors' earlier investigations on subgroups of algebraic number fields and division algebras [Pac. J. Math. 109, 165- 177 (1983; Zbl 0533.16009), Lect. Notes Math. 1006, 49-96 (1983; Zbl 0515.12006); Houston J. Math. 17, 207-226 (1991)]. Given an algebraic number field F whose ring of algebraic integers is \({\mathcal O}_ F\), let I(F) denote the set of all \({\mathcal O}_ F\)-submodules A of F which contain 1 and have the property that their quasi-endomorphism ring consists of the left multiplications by elements in F. Those members of I(F) which are closed under multiplication constitute a set of subrings of F which is denoted by E(F). The authors investigate which subsets \(\Xi\) of \(\Pi\) are carriers of groups in I(F) and which \(\Xi\) are carriers of rings in E(F). Their main results are: (1) a set \(\Xi\) of rational primes is the carrier of a group in I(F) if and only if \(\Xi\) is the carrier of a ring in E(F); and (2) in the case that F is a normal extension of \({\mathbb{Q}}\), the structure of the Galois group of F over \({\mathbb{Q}}\) determines, up to finitely many primes, those subsets of \(\Pi\) which are carriers of groups in I(F).
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 4212327
zbMATH Keywords: carrier
zbMATH Keywords: torsion-free abelian group
zbMATH Keywords: subgroups of algebraic number fields
zbMATH Keywords: quasi-endomorphism ring
zbMATH Keywords: carriers of groups
zbMATH Keywords: carriers of rings
zbMATH Keywords: Galois group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4212327