Torsion-free Abelian groups defined by an integral matrix.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.20043
publication date: 2012-09-12T00:00:00Z
full work available at URL: index.html
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 15B36
zbMATH DE Number: 6081654
zbMATH Keywords: torsion-free Abelian groups of finite rank
zbMATH Keywords: injective endomorphisms
zbMATH Keywords: direct limits
zbMATH Keywords: algebraic integers
zbMATH Keywords: dimension groups
zbMATH Keywords: almost completely decomposable groups
zbMATH Keywords: critical types
author: Manfred Dugas
MaRDI profile type: Publication
title: Torsion-free Abelian groups defined by an integral matrix.
published in: International Journal of Algebra
review text: Let \(\varphi\) be an injective endomorphism of a free Abelian group \(F\) of finite rank \(k\). Then the group \(G(\varphi,F)=\bigcup_{n\geq 0}\varphi^{-n}(F)\) is a torsion-free group of rank \(k\).NEWLINENEWLINE The author investigates the properties of such groups in terms of the eigenvalues of \(\varphi\) in the algebraic closure of \(\mathbb Q\). His paper contains many novel results on the structure of tffr groups, of which the following is the most remarkable:NEWLINENEWLINE Let \(G\) be a torsion-free group of rank \(k\). Then \(G\cong G(\varphi,F)\) for a free group \(F\) of rank \(k\) and an injective diagonalizable endomorphism \(\varphi\) of \(F\) if and only if \(G\) is almost completely decomposable and all the critical types of \(G\) are of ring type and divisible by only finitely many primes.
reviewed by: Phillip Schultz
description: scientific article; zbMATH DE number 6081654