On the homological significance of the group of rationals with square- free denominators
type: Item
instance of: scholarly article
title: On the homological significance of the group of rationals with square- free denominators
zbMATH Open document ID: 0638.20031
DOI: 10.1007/BF01191264
author: Werner Krandick
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: Frattini subgroups and Frattini closures of Abelian groups were studied by L. Fuchs and V. Dlab as early as in 1960. Now it turns out that for torsion-free Abelian groups these constructions are obtained quite easily by applying the functors \(\text{Hom}(R,A)\) and \(R \otimes A\), respectively, where \(R\) denotes the group of rationals with square-free denominators. Furthermore, the group \(R\) is closely related to the natural diagonal homomorphism \(\Delta_A : A \to \prod_{(p\text{ prime})}A / pA\), for \(A\) Abelian. This homomorphism is epi iff \(\text{Ext}(R,A) = 0\). In particular, \(\text{Ext}(R,A) = 0\), if the set \(\Sigma(A)\) of primes \(p\) with \(A/pA \neq 0\) is finite. Conversely, it is shown that for Frattini closures A finiteness of \(\Sigma(A)\) is equivalent to the vanishing of \(\text{Ext}(R,A)\).
Mathematics Subject Classification ID: 20K40
Mathematics Subject Classification ID: 20K35
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 18G15
zbMATH DE Number: 4040180
zbMATH Keywords: Frattini subgroups
zbMATH Keywords: Frattini closures
zbMATH Keywords: torsion-free Abelian groups
zbMATH Keywords: vanishing of Ext
MaRDI profile type: Publication
cites work: Q3269176
cites work: Q5609457
cites work: Q3861607
cites work: On the values of the functor \(\lim^1\)
full work available at URL: https://doi.org/10.1007/bf01191264
OpenAlex ID: W2083895140
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description: scientific article; zbMATH DE number 4040180