Kurosch invariants for torsion-free modules over Nagata valuation domains
type: Item
instance of: scholarly article
title: Kurosch invariants for torsion-free modules over Nagata valuation domains
zbMATH Open document ID: 0781.13004
DOI: 10.1016/0022-4049(92)90120-5
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: For an integral domain \(R\), \(\text{fr}(R)=n\) means that there exists an indecomposable torsion-free \(R\)-module of rank \(n\) and every torsion-free \(R\)-module of finite rank greater than \(n\) is decomposable. In the present paper the author investigates \(\text{fr}(R)\) for a non-complete discrete rank one valuation domain \(R\) (i.e. Nagata valuation domain) and direct decompositions of torsion-free \(R\)-modules. Let \(R^*\) be the completion of \(R\) and let \(Q\), \(Q^*\) be the respective fields of quotients. The author examines the case when \([Q^*:Q]=2\) or 3. In the case of degree 2 he proves a final result that \(\text{fr}(R)=2\) (theorem 8). For the proof he uses Kurosch matrix invariants of torsion-free \(R\)-modules and some relationships between decomposition of torsion-free \(R\)-module and decomposition of a representative matrix. In the case of \([Q^*:Q]=3\) he obtains several results on direct decompositions of torsion-free modules (for proofs he uses also Kurosch invariants). For example, if \(R\) is a Nagata valuation domain with \([Q^*:Q]=3\) and \(A\) is a torsion-free \(R\)-module of rank 4 then \(A\) is decomposable (theorem 15).
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 166143
zbMATH Keywords: direct decompositions of torsion-free modules
zbMATH Keywords: integral domain
zbMATH Keywords: Nagata valuation domain
zbMATH Keywords: Kurosch matrix invariants
author: P. Zanardo
reviewed by: Jiří Močkoř
MaRDI profile type: Publication
full work available at URL: 0022-4049(92)90120-5
OpenAlex ID: W1964930762
cites work: A Duality for Torsion-Free Modules of Finite Rank Over a Discrete Valuation Ring
cites work: Discrete valuation domains and ranks of their maximal extensions
cites work: Quasi-Direct Decompositions of Torsion-Free Abelian Groups of Infinite Rank.
cites work: Polyserial modules over valuation domains
cites work: Q5661989
cites work: On the analogue of Corner's finite rank theorem for modules over valuation domains
cites work: Q5829819
cites work: Modules Over Dedekind Rings and Valuation Rings
cites work: The Decomposability of Torsion Free Modules of Finite Rank
cites work: Q4052239
cites work: Q5735274
cites work: On the completion of a valuation ring
cites work: Rank-two torsion-free modules over valuation domains
cites work: Decomposition Problems for Modules Over Valuation Domains
description: scientific article; zbMATH DE number 166143