The flat dimension of mixed abelian groups as \(E\)-modules
type: Item
instance of: scholarly article
title: The flat dimension of mixed abelian groups as \(E\)-modules
zbMATH Open document ID: 0843.20045
author: H. Pat Goeters
author: W. J. Wickless
author: Ulrich F. Albrecht
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-08-13T00:00:00Z
review text: Let \(G\) be an abelian group and let \(E = E(G)\) denote its endomorphism ring. Then \(G\) is in a natural way a left \(E\)-module. It was R. S. Pierce who asked for a way to calculate the flat dimension \(fd_E(G)\) of the \(E\)-module \(_EG\). The present paper is devoted to this topic for the case that \(G\) belongs to a certain class \(\mathcal G\) of mixed abelian groups which was introduced by \textit{S. Glaz} and \textit{W. Wickless} [Commun. Algebra 22, No. 4, 1161-1176 (1994; Zbl 0801.20037)]. Let \(G \in {\mathcal G}\) and let \(T\) denote the maximal divisible subgroup of \(G\). Then \(G/T\) is a vector space over the field \(\mathbb{Q}\) of rational numbers of some finite dimension \(n\) which gives rise to a ring homomorphism \(\mu\) from \(E(G)\) into the ring \(M_n(\mathbb{Q})\) of \(n \times n\) rational matrices. Let \(A\) denote the image of \(\mu\). Then \(A\) is a subalgebra of \(M_n(\mathbb{Q})\) which is unique up to isomorphism and is said to be realized by \(G\). Both \(G/T\) and \(M_n(\mathbb{Q})\) may be regarded as \(A\)-modules. The authors show that, for any \(G \in {\mathcal G}\), \(fd_E(G) = fd_A (G/T)\) and relate the projective dimension of the \(E\)-module \(G\) to the flat dimension of the \(A\)-module \(G/T\). They prove that, given any \(k\) with \(0 \leq k \leq \infty\), there exists \(G \in {\mathcal G}\) such that \(fd_E(G) = k\). The authors consider the question which subalgebras of \(M_n(\mathbb{Q})\) are realizable by some \(G \in {\mathcal G}\) and show that this class is fairly large. An example is given of a subalgebra of \(M_4(\mathbb{Q})\) which is not realizable.
reviewed by: Jutta Hausen
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 796890
zbMATH Keywords: endomorphism rings
zbMATH Keywords: left \(E\)-modules
zbMATH Keywords: flat dimension
zbMATH Keywords: mixed Abelian groups
zbMATH Keywords: maximal divisible subgroups
zbMATH Keywords: \(n\times n\) rational matrices
zbMATH Keywords: projective dimension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072238
OpenAlex ID: W2010222445
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DOI: 10.1216/RMJM/1181072238
description: scientific article; zbMATH DE number 796890