Localizations of torsion-free Abelian groups. II.
type: Item
instance of: scholarly article
title: Localizations of torsion-free Abelian groups. II.
zbMATH Open document ID: 1082.20031
author: Manfred Dugas
published in: Journal of Algebra
publication date: 2005-04-07T00:00:00Z
review text: A homomorphism \(\alpha\colon A\to B\) between Abelian groups \(A,B\) is called a localization of \(A\) if every homomorphism \(\varphi\) from \(A\) to \(B\) has a unique extension to an endomorphism \(\psi\) of \(B\) in the sense that \(\varphi=\psi\circ\alpha\). More generally, the author calls an Abelian group \(C\) perpendicular to \(\alpha\) if for every homomorphism \(\varphi\) from \(A\) to \(C\) there is a unique \(\psi\) from \(B\) to \(C\) such that \(\varphi=\psi\circ\alpha\). Hence \(B\) is perpendicular to \(\alpha\) if and only if \(\alpha\) is a localization of \(A\) and in this case \(B\) is called a localization of \(A\). Each localization gives rise to an idempotent localization functor \(L_\alpha\) from the category of Abelian groups into the reflective subcategory of all Abelian groups perpendicular to \(\alpha\). Furthermore, there is a canonical homomorphism \(\eta_Y\colon Y\to L_\alpha(Y)\) such that \(\eta_Y\) is perpendicular to \(X\) for all \(X\) which are perpendicular to \(\alpha\) and \(\eta_A=\alpha\). This homomorphism \(\eta_Y\) is called the coaugmentation map of \(Y\). Localizations arose in general category theory and have received a lot of attention recently, especially in the category of Abelian groups. For instance it was shown that localizations of the group of integers are exactly the so-called \(E\)-rings which had been introduced by Schultz. In this paper the author studies localizations \(\alpha\colon L\to M\) where \(L\) is a subgroup of the additive group of rational numbers \(\mathbb{Q}\) and \(M\) is torsion-free. It is proved that there are drastic differences depending on whether or not \(L\) is a subring of \(\mathbb{Q}\). If \(L\) is a ring, then \(M\) is simply an \(E\)-ring which is also an \(L\)-module. In this case the author describes the localization functor explicitly. If \(L\) is not a subring but \(M=M(\tau)\) where \(\tau\) is the type of \(L\), then there exists an \(E\)-ring \(H\) which is an \(\text{En…
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 2153327
zbMATH Keywords: \(E\)-rings
zbMATH Keywords: localizations
zbMATH Keywords: torsion-free Abelian groups
zbMATH Keywords: homomorphisms
zbMATH Keywords: endomorphisms
zbMATH Keywords: idempotent localization functors
zbMATH Keywords: categories of Abelian groups
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.11.013
OpenAlex ID: W4210744982
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DOI: 10.1016/J.JALGEBRA.2004.11.013
description: scientific article; zbMATH DE number 2153327