Does the automorphism group generate the endomorphism ring in \(\text{Rep}(S,R)\)?
type: Item
instance of: scholarly article
title: Does the automorphism group generate the endomorphism ring in \(\text{Rep}(S,R)\)?
zbMATH Open document ID: 0972.16019
published in: Journal of Algebra
publication date: 2001-08-23T00:00:00Z
review text: In 1963 Laszlo Fuchs posed the question of when the automorphism group of an Abelian group additively generates the endomorphism ring. Since then Fuchs's problem and related generalizations have been extensively studied in categories such as Abelian groups, vector spaces, free modules over PID's, general rings, and Warfield modules. The current work investigates this topic in the setting of representations of a commutative ring \(R\) over a finite partially ordered set \((S,\leq)\). The question is answered fully when \(R\) is a field (\(\not=\text{GF}(2)\)) and partial results are obtained when \(R\) is a principal ideal domain which is not a field. An element \(\underline V\) of the category \(\text{Rep}(S,R)\) is a family \(\underline V=(V,V_i\mid i\in S)\) where \(V\) is a free \(R\)-module and each \(V_i\) is a free \(R\)-submodule of \(V\) such that \(V_i\subseteq V_j\) whenever \(i\leq j\). A homomorphism \(\underline\phi\) from \(\underline V\) to \(\underline W\) is a homomorphism \(\phi\colon V\to W\) such that \(\phi|_{V_i}\) maps \(V_i\) to \(W_i\). One of Strüngmann's foremost results states that when \(R=k\) is a field (\(\not=\text{GF}(2)\)) and either \(V\) is a finite dimensional vector space over \(k\) or \(|S|\leq 3\), then \(\text{End}_k(\underline V)=\Aut_k(\underline V)+\Aut_k(\underline V)\). When \(|S|>3\), the author demonstrates that anything can happen with regard to Fuchs's property.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 1525270
zbMATH Keywords: automorphism groups
zbMATH Keywords: endomorphism rings
zbMATH Keywords: representations
zbMATH Keywords: finite posets
zbMATH Keywords: \(m\)-sum property
author: Lutz Strüngmann
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
full work available at URL: jabr.2000.8361
OpenAlex ID: W2007012668
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DOI: 10.1006/JABR.2000.8361
description: scientific article; zbMATH DE number 1525270