Almost flat abelian groups
type: Item
instance of: scholarly article
title: Almost flat abelian groups
zbMATH Open document ID: 0847.20052
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-09-30T00:00:00Z
review text: In a series of lectures at the University of Connecticut at Storrs R. S. Pierce raised in 1989 several questions concerning the impact of flatness of an abelian group \(A\) as a left module over its endomorphism ring \(E(A)\) on \(A\) as group, or \(\mathbb{Q}\otimes_\mathbb{Z} A\) as \(\mathbb{Q}\otimes E(A)\)-module. These questions are answered in the paper under discussion. The investigations lead the authors to say that \(A\) is almost flat as \(E(A)\)-module if \(\text{Tor}^1_{E(A)}(M,A)\) is a bounded abelian group for all right \(E(A)\)-modules \(M\). The class of almost flat \(E(A)\)-modules is shown to be closed under quasi-isomorphism. This theorem seems to be the modified answer intended by Pierce after ruling out counter Example 2.9. It follow characterizations of almost flat \(E(A)\)-modules in case \(A\) has finite rank which are interesting on their own.
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 878056
zbMATH Keywords: almost flat modules
zbMATH Keywords: endomorphism rings
zbMATH Keywords: bounded Abelian groups
zbMATH Keywords: quasi-isomorphisms
author: Ulrich F. Albrecht
author: Pat Goeters
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072189
OpenAlex ID: W2065480224
cites work: Baer's Lemma and Fuchs' Problem 84a
cites work: Endomorphism rings of faithfully flat Abelian groups
cites work: Finite rank torsion free Abelian groups and rings
cites work: Examples of torsion-free groups flat as modules over their endomorphism rings
cites work: Generating cotorsion theories and injective classes
cites work: FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS
cites work: Q3204678
cites work: Extensions of torsion-free Abelian groups of finite rank
DOI: 10.1216/RMJM/1181072189
description: scientific article; zbMATH DE number 878056