Rank-two torsion-free modules over valuation domains
type: Item
instance of: scholarly article
title: Rank-two torsion-free modules over valuation domains
zbMATH Open document ID: 0685.13006
author: P. Zanardo
author: Luigi Salce
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108116
review text: This paper represents the first step in developing the theory of torsion free R-modules of finite rank for general valuation rings R symmetrically to the theory of finitely generated torsion R-modules. This symmetry is known for some special cases of R (e.g. for maximal valuation domains), on the other hand this symmetry is lost if R is almost maximal which fails to be maximal. The principal role in the paper have the class \({\mathcal F}_ 2(R)\) of indecomposable torsion-free R-modules of rank two and the class \({\mathcal G}_ 2(R)\) of two-generated indecomposable torsion R-modules. The authors present a universal construction of indecomposable torsion-free R-modules M of rank two (R non almost maximal) using a unit u of a maximal immediate extension S of R theorem 3.1). This construction is connected with compatible triples (L,H,I) of submodules of the quotient field of R, where L is isomorphic to a basic submodule B of M, H is isomorphic to M/B and \(I=\{a\in R: u\not\in aR+S\}\) is the so-called breath ideal of u. The authors receive then theorem 5.3, which characterizes isomorphic elements of \({\mathcal F}_ 2(R)\) and subsequently they receive a complete and independent set of invariants for \({\mathcal F}_ 2(R)\) in a form of some equivalence classes \([L,H,u]\). Then they show that symmetrical results hold for \({\mathcal G}_ 2(R)\), too.
reviewed by: Jiří Močkoř
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13C12
zbMATH DE Number: 4122096
zbMATH Keywords: rank-two torsion-free modules over valuation domains
zbMATH Keywords: breath ideal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4122096