\(h\)-divisible and cotorsion modules over one-dimensional Gorenstein rings
type: Item
instance of: scholarly article
title: \(h\)-divisible and cotorsion modules over one-dimensional Gorenstein rings
zbMATH Open document ID: 0795.13013
author: E. Enochs
author: Overtoun M. G. Jenda
published in: Journal of Algebra
publication date: 1994-09-14T00:00:00Z
review text: The aim of this paper is to study the class \(\mathcal C\) of all commutative Noetherian rings whose self injective dimension (id) is at most one. Results from Dedekind rings are extended in this frame. Let \(R\) be a commutative Noetherian ring and \(M\) an \(R\)-module. \(M\) is \(h\)-divisible if it is a homomorphic image of an injective \(R\)-module. A characterization of \(\mathcal C\) in terms of \(h\)-divisibility is given. \(M\) is said to be cotorsion if \(\text{Ext}_ R^ 1(F,M)=0\) for all flat \(R\)-modules \(F\). If \(R\in{\mathcal C}\) then \(M\) is cotorsion if and only if \(\text{Ext}_ R^ 1(E(R),M)=0\), \(E(R)\) being the injective envelope of \(R\). If \(R\in{\mathcal C}\) is indecomposable and \(F\) is a flat \(R\)-module then \(\text{Ext}^ 1_ R(E(R),F)\) is the completion of \(F\) with respect to the adic topology of all maximal ideals of \(R\). An extension of Harrison duality [\textit{D. K. Harrison}, Ann. Math., II. Ser. 69, 366-391 (1959; Zbl 0100.02901)] for injective torsion \(R\)-modules when \(R\in{\mathcal C}\) is given.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13C12
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 554251
zbMATH Keywords: cotorsion module
zbMATH Keywords: self injective dimension
zbMATH Keywords: Harrison duality
MaRDI profile type: Publication
full work available at URL: jabr.1993.1228
OpenAlex ID: W1986772414
DOI: 10.1006/JABR.1993.1228
description: scientific article; zbMATH DE number 554251