An Artin problem for division ring extensions and the pure semisimplicity conjecture. I
type: Item
instance of: scholarly article
title: An Artin problem for division ring extensions and the pure semisimplicity conjecture. I
zbMATH Open document ID: 0873.16010
DOI: 10.1007/BF01273341
author: Daniel Simson
published in: Archiv der Mathematik
publication date: 1997-11-02T00:00:00Z
review text: A ring \(R\) is right pure semisimple if every right \(R\)-module is a direct sum of finitely generated modules and it is of finite representation type if, further, there are just finitely many indecomposable modules up to isomorphism. The pure semisimplicity conjecture is that every right pure semisimple ring is of finite representation type. The author shows that the following would provide a counterexample to the pure semisimplicity conjecture: division rings \(F\subseteq G\) such that \(\dim(_FG)\) is infinite, \(\dim(G_F)=2\) and such that the right dimension of each of the iterated right dualisations of \(_FG_G\) with respect to \(\Hom_G(_F(-)_G,G)\) is 2. If there exist such \(F\subseteq G\) then the hereditary ring \(R_G=\left(\begin{smallmatrix} F & _FG_G \\ 0 & G\end{smallmatrix}\right)\) would have the following properties, among others: \(R_G\) would be right pure semisimple; the intersection of the finite powers of the radical of \(\text{mod-}R_G\) would be non-zero (and hence \(R_G\) would not be of finite representation type) but its square would be zero; for each \(m\geq 2\), the number of isomorphism classes of indecomposable right \(R_G\)-modules of length \(m\) would be 0 or 1.
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 937055
zbMATH Keywords: indecomposable right modules
zbMATH Keywords: direct sums of finitely generated modules
zbMATH Keywords: finite representation type
zbMATH Keywords: pure semisimplicity conjecture
zbMATH Keywords: right pure semisimple rings
zbMATH Keywords: division rings
zbMATH Keywords: hereditary rings
Wikidata QID: Q122932086
reviewed by: Mike Prest
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 937055