Warfield domains
type: Item
instance of: scholarly article
title: Warfield domains
zbMATH Open document ID: 0873.13020
published in: Journal of Algebra
publication date: 1997-11-04T00:00:00Z
review text: Warfield domains are commutative integral domains \(R\) for which, given any \(R\)-submodule \(A\) of the field of quotient \(Q\) of \(R\), all \(A\)-torsionless \(\text{End}_R (A)\)-modules of finite rank are \(A\)-reflexive. This notion has evolved from investigations of R. A. Warfield on abelian groups. The article under review aims at characterizing more generally Warfield domains, since recently results have been obtained for Dedekind and later on for special noetherian integral domains. The extended results read as follows: (1) For a noetherian domain every ideal is two generated if and only if it is a Warfield domain. (2) If \(R\) is integrally closed, then \(R\) is a Warfield domain if and only if \(R\) is an almost maximal strongly discrete Prüfer domain. The proofs use the important characterization of Warfield domains as totally reflexive domains (theorem 6.6). For its proof plenty of results concerning \(A\)-reflexive, \(A\)-divisorial and local divisorial domains are necessary.
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 954785
zbMATH Keywords: two generated ideal
zbMATH Keywords: Warfield domains
zbMATH Keywords: Prüfer domain
zbMATH Keywords: totally reflexive domains
Wikidata QID: Q105710879
author: Silvana Bazzoni
author: Luigi Salce
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: jabr.1996.0353
OpenAlex ID: W4212915462
DOI: 10.1006/JABR.1996.0353
description: scientific article; zbMATH DE number 954785