Injective envelopes and flat covers of Matlis reflexive modules
type: Item
instance of: scholarly article
title: Injective envelopes and flat covers of Matlis reflexive modules
zbMATH Open document ID: 0756.13007
DOI: 10.1016/0022-4049(92)90050-P
author: Richard G. Belshoff
author: Jin Zhong Xu
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: All rings in this paper are commutative with identity and noetherian. For a local ring \((R,{\mathfrak m})\), the Matlis dual \(\text{Hom}_ R(M,E)\), where \(E\) is the injective envelope \(E_ R(R/{\mathfrak m})\), is denoted by \(M^ \lor\). If the canonical map \(M\to M^{\lor\lor}\) is an isomorphism, then \(M\) is said to be (Matlis) reflexive. The relation between the reflexivity of a module \(M\) and its injective hull \(E_ R(M)\) is studied for different types of rings and modules. Amongst these results, the most important one states that if \((R,{\mathfrak m})\) is a complete local ring with \(\dim R\leq 1\), then an \(R\)-module \(M\) is reflexive if and only if \(E(M)\) is reflexive. The paper culminates in the main result in which the authors show that for a commutative noetherian local ring \(R\), every reflexive \(R\)-module has a reflexive injective envelope if and only if every reflexive \(R\)-module has a reflexive flat cover. This occurs if and only if \(R\) is complete and has Krull dimension less than or equal to 1.
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 61223
zbMATH Keywords: Matlis reflexive module
zbMATH Keywords: complete ring
zbMATH Keywords: injective hull
zbMATH Keywords: noetherian local ring
zbMATH Keywords: reflexive injective envelope
zbMATH Keywords: reflexive flat cover
Wikidata QID: Q114215379
reviewed by: Frieda Theron
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 61223