Generating cotorsion theories and injective classes
type: Item
instance of: scholarly article
title: Generating cotorsion theories and injective classes
zbMATH Open document ID: 0643.20033
DOI: 10.1007/BF01903623
author: B. George
published in: Acta Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: Let H be a class of torsion-free abelian groups of finite rank (hereafter, simply ``groups''). This paper considers three classes containing H; (1) Let \(P(H)\) (respectively, \(I(H)\)) be the class of all groups Y satisfying \(Ext(Y,X)=0\) (respectively, \(Ext(X,Y)=0)\) for all X in H. (2) Let \(C(H)\) be the class of groups A such that each X in H is injective with respect to every pure embedding \(A\to B\) of groups. Define \(IC(H)\) to be the class of all groups injective with respect to each pure embedding \(A\to B\) of groups with A in C(H). (3) Let \(E(H)\) be the class of all exact sequences of groups relative to which each X in H is injective and let \(IE(H)\) be the class of all groups Y such that Y is injective with respect to each sequence in \(E(H)\). Clearly, \(H\subset I(P(H)) \cap IC(H) \cap IE(H)\). In fact, \(IE(H)\subset IC(H) \cap I(P(H))\). The author proves a number of other results about these classes, including the following characterization of IP(H). Theorem 4. The class \(IP(H)\) is the class of all groups Y satisfying (a) \(\{p\) prime \(|\;pY\neq Y\}\subset \cup_{X\in H}\{p|\;pX\neq X\}\) and (b) If J is an infinite subset of \(\{p|\) \(pY\neq Y\}\), then there is an X in H with \(J\cap \{p|\) \(pX\neq X\}\) infinite.
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 4047957
zbMATH Keywords: cotorsion theory
zbMATH Keywords: injective class
zbMATH Keywords: torsion-free abelian groups of finite rank
zbMATH Keywords: pure embedding
zbMATH Keywords: exact sequences
reviewed by: Charles Vinsonhaler
MaRDI profile type: Publication
Wikidata QID: Q114233902
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full work available at URL: https://doi.org/10.1007/bf01903623
OpenAlex ID: W2152128000
description: scientific article; zbMATH DE number 4047957