Prime submodules, the descent invariant, and modules of finite length
type: Item
instance of: scholarly article
title: Prime submodules, the descent invariant, and modules of finite length
zbMATH Open document ID: 0878.13005
published in: Journal of Algebra
publication date: 1998-01-04T00:00:00Z
review text: The authors extend the notion of prime ideals to modules over commutative rings and try to develop a (local) dimension theory for modules. As the first step, a class of particularly simple prime submodules, the so-called 0-submodules, is investigated in section 2. A submodule is defined to be a 0-submodule if the factor module is torsion-free. It is proved that over regular local rings a 0-submodule of rank 1 of a finitely generated free module is always free. This result is used in theorem 2.6 to characterize vector bundles of rank \(r-1\), generated by \(r\) elements over the punctured spectrum. The prime dimension of a module is introduced and investigated in section 3. This careful investigation leads to the notion of descent which is used to characterize reflexive modules over regular local rings by combinatorial invariants in section 4. Finitely generated modules over regular local rings satisfying Serre's condition \(S_n\) are also discussed. In section 5 finitely generated reflexive modules over local domains are characterized by using matrices and syzygies. Torsion-free finitely generated modules over polynomial rings with two variables over algebraically closed fields are discussed in section 6.
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 1013986
zbMATH Keywords: prime dimension of a module
zbMATH Keywords: finitely generated reflexive modules
author: Agustin Marcelo
author: Jaime Muñoz Masqué
reviewed by: Ánh Pham Ngoc
MaRDI profile type: Publication
full work available at URL: jabr.1996.6871
OpenAlex ID: W2072999380
cites work: Q4274969
cites work: Q3686801
cites work: Stable reflexive sheaves
cites work: The dimension and components of symmetric algebras
cites work: Q4295730
cites work: Q4293985
DOI: 10.1006/JABR.1996.6871
description: scientific article; zbMATH DE number 1013986