Testing for cotorsionness over domains
type: Item
instance of: scholarly article
title: Testing for cotorsionness over domains
zbMATH Open document ID: 1167.13303
author: Rüdiger Göbel
author: László Fuchs
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2008-12-01T00:00:00Z
full work available at URL: https://eudml.org/doc/108731
review text: Summary: We are looking for reduced modules over an integral domain to test the cotorsionness of modules. Our focus is on the cotorsion theories due to Matlis, Enochs and Warfield, respectively. If \(\kappa\) is a cardinal such that \(\kappa^{\aleph_0}=2^{\kappa}\), then for domains \(R\) for which \(Q/R\) (\(Q\) is the quotient field of \(R\)) is not self-small there is a reduced strongly flat \(R\)-module \(U_k\) of rank \(2^\kappa\) that can be used to test whether or not an \(R\)-module of cardinality \(2^k\) is Matlis cotorsion; however, no such module exists if the cardinality restriction is removed. We also establish the existence of reduced modules which test Matlis cotorsion modules for Enochs or Warfield cotorsionness. For a given torsion-free or flat \(R\)-module \(A\) of cardinality \(\kappa\), we construct an Enochs, resp. Matlis cotorsion \(R\)-module \(M\) of cardinality \(2^\kappa\) with \(\text{Ext}^1_R (A; M)=0\) that is not Warfield, resp. not Enochs cotorsion.
Mathematics Subject Classification ID: 13C12
zbMATH DE Number: 5376048
MaRDI profile type: Publication
cites work: An independence result on cotorsion theories over valuation domains
cites work: On strongly flat modules over integral domains
cites work: ALL MODULES HAVE FLAT COVERS
cites work: Q3995749
cites work: How To Make Ext Vanish
cites work: Relative homological algebra
cites work: Q4513600
cites work: Q4466609
cites work: Q3356537
cites work: Independence in Completions and Endomorphism Algebras
cites work: Solution of two problems on cotorsion abelian groups
cites work: Approximations and endomorphism algebras of modules.
cites work: Infinite Abelian groups and homological methods
cites work: Balanced Subgroups of Abelian Groups
cites work: Q3852172
cites work: On divisible modules over domains
cites work: Cotorsion modules
cites work: Q3861607
description: scientific article; zbMATH DE number 5376048