The Jordan-Zassenhaus theorem and direct decompositions
type: Item
instance of: scholarly article
title: The Jordan-Zassenhaus theorem and direct decompositions
zbMATH Open document ID: 0964.16017
published in: Journal of Algebra
publication date: 2001-05-02T00:00:00Z
review text: Recall, that the Jordan-Zassenhaus theorem states that, if \(R\) is a Dedekind domain whose field \(Q\) of quotients is a global field, then for each \(R\)-order \(S\) in a semisimple algebra over \(Q\), and for each positive integer \(n\), there are only finitely many isomorphism classes of left \(S\)-lattices of rank \(\leq n\). For a Dedekind domain \(R\) of characteristic 0 the Jordan-Zassenhaus theorem holds if and only if for every non-zero ideal \(I\) of \(R\), the factor-ring \(R/I\) is finite, and every finite rank torsion-free \(R\)-module has but a finite number of non-isomorphic direct summands (Theorem 5.2). For a Dedekind \(R\) of characteristic 0 the following are equivalent: (i) every torsion-free \(R\)-module of finite rank has, up to equivalence, but a finite number of direct decompositions; (ii) for every \(R\)-order \(S\), the class \(\text{Cl}(S)\) is finite, and torsion-free \(R\)-modules of finite rank have the almost cancellation property; (iii) for every \(R\)-order \(S\), the genus classes of \(S\)-lattices are finite; (iv) every \(R\)-order \(S\) satisfies: (a) for every ideal \(I\) of \(S\) with \(I\cap R\neq 0\), the subgroup \(\overline{U(S)}\) has finite index in \(U(S/I)\) and (b) there is \(0\neq r\in R\) such that every \(S\)-lattice in the genus of \(S\) is \(S\)-isomorphic to an \(S\)-submodule between \(rS\) and \(S\); (v) every \(R\)-order \(S\) satisfies: (a) for every ideal \(I\) of \(S\) with \(I\cap R\neq 0\), the subgroup \(\overline{U(S)}\) has finite index in \(U(S/I)\) and (b) the genus of \(S\) is finite (Theorems 5.1 and 4.3).
reviewed by: Ladislav Bican
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 1512989
zbMATH Keywords: Dedekind domains
zbMATH Keywords: orders
zbMATH Keywords: lattices
zbMATH Keywords: almost cancellation property
zbMATH Keywords: genus
zbMATH Keywords: semisimple algebras
zbMATH Keywords: torsion-free modules
zbMATH Keywords: direct summands
zbMATH Keywords: direct decompositions
author: László Fuchs
author: P. Vámos
MaRDI profile type: Publication
full work available at URL: jabr.2000.8416
OpenAlex ID: W2008194377
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DOI: 10.1006/JABR.2000.8416
description: scientific article; zbMATH DE number 1512989