Endomorphism rings of abelian groups as isomorphic restrictions of full endomorphism rings
type: Item
instance of: scholarly article
title: Endomorphism rings of abelian groups as isomorphic restrictions of full endomorphism rings
zbMATH Open document ID: 0792.20054
DOI: 10.1007/BF02330871
published in: Periodica Mathematica Hungarica
publication date: 1993-02-08T00:00:00Z
review text: An abelian group is torsion-free if all its elements except zero are of infinite order. An associative ring with identity is called torsion-free if its additive group is torsion-free. The author, with the aid of Corner's technique, proves the following theorem: Let \(G\) be a countable, reduced, torsion-free group. Let \(R\) be a subring of the endomorphism ring \(E({G})\) of \(G\) which contains the identity of \(E({G})\). Then there exists a reduced, torsion-free group \(H\) whose endomorphism ring \(E(H)\) is isomorphic to \(R\), moreover, \(G\) is a pure, fully invariant subgroup of \(H\) and \(R\) is a ``faithful'' restriction of \(E(H)\) to \(G\), that is \(E(H)|_{G} = R\) and \(\varphi \to \varphi\mid_{G}\) is an isomorphism of \(E(H)\) to \(R\). Consequence. Every countable, reduced, torsion-free group \(G\) can be embedded in a countable, reduced, torsion- free group \(H\) which has only trivial endomorphisms (a trivial endomorphism is the multiplier operation of a fixed integer).
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 118322
zbMATH Keywords: associative ring
zbMATH Keywords: additive group
zbMATH Keywords: countable, reduced, torsion-free group
zbMATH Keywords: endomorphism ring
zbMATH Keywords: pure, fully invariant subgroup
author: Nguyen Thang Dao
reviewed by: I. Kh. Bekker
MaRDI profile type: Publication
cites work: Every Countable Reduced Torsion-Free Ring is an Endomorphism Ring
cites work: Three examples on hopficity in torsion-free abelian groups
cites work: Q5609457
cites work: Q5581748
full work available at URL: https://doi.org/10.1007/bf02330871
OpenAlex ID: W2618192011
description: scientific article; zbMATH DE number 118322