Reflexive D-rings
type: Item
instance of: scholarly article
title: Reflexive D-rings
zbMATH Open document ID: 0687.13013
DOI: 10.1016/0021-8693(88)90184-6
author: David E. Rush
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: A domain is called reflexive if every torsionless module of finite rank is reflexive, and D-rings R are domains with the following property: Every torsion-free R-module of finite rank is a direct sum of rank 1 modules. Matlis characterized several classes of D-rings, among them the integrally closed and the Noetherian ones. A general characterization of D-rings is lacking however. It follows from Matlis' results that every quasi-local D-ring is an intersection of quasi-local reflexive D-rings contained in its integral closure. This leads the author to study quasi- local reflexive D-rings. In the following Q denotes the field of fractions of R. We quote some of the results obtained: (1) A quasi-local D-ring is reflexive if and only if each ideal is reflexive. (2) Let R be quasi- local, V its integral closure, and I the conductor; then R is reflexive if and only if V and R/I are maximal valuation rings, Q is a linearly compact R-module, and all ideals of R are reflexive. (3) With the notations of (2), R is a reflexive D-ring if and only if V and R/I are maximal valuations rings and Q/R is the injective envelope of the residue class field of R.
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 4125524
zbMATH Keywords: quasi-local reflexive D-rings
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90184-6
OpenAlex ID: W1980511499
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description: scientific article; zbMATH DE number 4125524