Slenderness, completions, and duality for primary Abelian groups
type: Item
instance of: scholarly article
title: Slenderness, completions, and duality for primary Abelian groups
zbMATH Open document ID: 0876.20033
published in: Journal of Algebra
publication date: 1997-10-16T00:00:00Z
review text: All groups are abelian here. Let \(A\) be a group, and \(E=\text{End }A\) its endomorphism ring. For any group \(G\), set \(G^*=\text{Hom}(G,A)\) and note that \(G^*\) is naturally a left \(E\)-module. On the other hand, if \(M\) is a left \(E\)-module, then \(M^*=\text{Hom}_E(M,A)\) is an abelian group. There is the natural evaluation map \(\sigma_G\colon G\to G^{**}\): \(\sigma_G(x)(f)=f(x)\) and \(G\) is called \(A\)-reflexive if \(\sigma_G\) is an isomorphism. A left \(E\)-module \(M\) is called \(E\)-slender if every \(E\)-homomorphism \(\prod_{j\in\omega}E_j\to M\), where \(E_j\cong E\), maps some submodule \(\prod_{j\geq m}E_j\) to 0. If \(G=\bigoplus_{j\in\omega}A_j\), where \(A_j\cong A\), then \(G^*=\prod_{j\in\omega}E_j\), and \(A\) is \(E\)-slender if and only if \(G\) is \(A\)-reflexive. This indicates that there is an interaction between reflexivity and slenderness. There is also a connection with topological completion. Let \(G\) be a group and \(\mathcal T\) a neighborhood base at 0 for a linear topology on \(G\). If \(A\) is a group with discrete topology such that \(G/X\) is \(A\)-reflexive for all \(X\in\mathcal T\), and further every homomorphism \(G\to A\) is continuous, then algebraically \(G^{**}\cong\widetilde G\), where \(\widetilde G\) denotes the completion of the topological group \((G,\mathcal T)\). These are the themes of the paper under discussion. Major results are the following. Theorem 1.6. Suppose that \(A\) is an unbounded reduced \(p\)-group and \(E=\text{End }A\). Then \(A\) is \(E\)-slender if and only if \(p^\omega A\) is a slender \(E\)-module and \(A/p^\omega A\) is not torsion-complete. A corollary answers affirmatively a conjecture of the reviewer: An unbounded separable \(p\)-group \(A\) is \(\text{End }A\)-slender if and only if \(A\) is not torsion-complete. Recall that a \(p\)-group \(G\) is \(p^{\omega+n}\)-projective if it contains a \(p^n\)-bounded subgroup \(P\) such that \(G/P\) is a direct sum of cyclic groups. I…
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K45
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 1014036
zbMATH Keywords: slenderness
zbMATH Keywords: duality
zbMATH Keywords: reflexivity
zbMATH Keywords: linear topology
zbMATH Keywords: completablity
zbMATH Keywords: direct sums of cyclics
zbMATH Keywords: endomorphism rings
zbMATH Keywords: topological completions
zbMATH Keywords: unbounded reduced \(p\)-groups
zbMATH Keywords: separable \(p\)-groups
zbMATH Keywords: \(p^ \omega\)-projective groups
author: Patrick W. Keef
reviewed by: Adolf Mader
MaRDI profile type: Publication
full work available at URL: jabr.1997.6797
OpenAlex ID: W1994358208
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DOI: 10.1006/JABR.1997.6797
description: scientific article; zbMATH DE number 1014036