On the Bassian property for abelian groups
type: Item
instance of: scholarly article
title: On the Bassian property for abelian groups
zbMATH Open document ID: 1497.20058
published in: Archiv der Mathematik
publication date: 2021-11-26T00:00:00Z
review text: The authors call a (non-trivial) abelian group \(G\) Bassian, if there is no injective homomorphism \(G\to G/H\), unless the subgroup \(H=0\). It appears that such a group has to have small rank, for this to happen, so as to limit possibilities for groups \(G/H\). Complete characterization of such groups is obtained by considering separately classes of divisible/reduced groups, torsion groups, torsion-free groups and genuinely mixed groups: A reduced abelian group is Bassian if and only if its torsion-free rank is finite and all the \(p\)-ranks are finite (for all primes \(p\)). A non-reduced abelian group is Bassian, if and only if its divisible part is a direct sum of a finite number of copies of the rationals and its reduced part is Bassian. Some elementary properties of Bassian groups are exhibited.
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 13E15
zbMATH DE Number: 7433916
zbMATH Keywords: abelian group
zbMATH Keywords: Hopfian group
zbMATH Keywords: Bassian group
zbMATH Keywords: rank
author: Andrey R. Chekhlov
author: Peter Danchev
author: B. Goldsmith
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: s00013-021-01655-4
OpenAlex ID: W3200705297
cites work: The flat dimension of mixed abelian groups as \(E\)-modules
cites work: Q3488297
cites work: A Finiteness Property of Affine Algebras
cites work: Q5609457
cites work: Q5673086
cites work: Abelian Groups
cites work: Regular and principal projective endomorphism rings of mixed abelian groups
DOI: 10.1007/S00013-021-01655-4
description: scientific article; zbMATH DE number 7433916