Injectivity relative to closed submodules.
type: Item
instance of: scholarly article
title: Injectivity relative to closed submodules.
zbMATH Open document ID: 1175.16006
author: Engin Mermut
author: Catarina Santa-Clara
author: Patrick F. Smith
published in: Journal of Algebra
publication date: 2009-04-21T00:00:00Z
review text: Let \(R\) be a ring. An \(R\)-module \(X\) is called \(c\)-injective if, for every closed submodule \(L\) of every \(R\)-module \(M\), every homomorphism from \(L\) to \(X\) lifts to \(M\). The following two results are obtained in this paper: (1) If \(R\) is a Dedekind domain, then an \(R\)-module \(X\) is \(c\)-injective iff \(X\) is isomorphic to a direct product of homogeneous semisimple \(R\)-modules and injective \(R\)-modules. (2) A commutative Noetherian domain \(R\) is Dedekind iff every simple \(R\)-module is \(c\)-injective.
reviewed by: Tong Wenting
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 5545847
zbMATH Keywords: injective modules
zbMATH Keywords: closed submodules
zbMATH Keywords: Dedekind domains
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.11.004
OpenAlex ID: W2094643766
Wikidata QID: Q122681190
cites work: Modules with decompositions that complement direct summands
cites work: Self-c-injective Abelian groups.
cites work: Q4843160
cites work: Multiplication modules
cites work: Q5609457
cites work: Q4513600
cites work: Q3994888
cites work: High extensions of Abelian groups
cites work: Q3233254
cites work: Modules Over Dedekind Rings and Valuation Rings
cites work: Injective Structures
cites work: Q3786586
cites work: Preface to the English translation
cites work: Q3995216
cites work: Group rings of polycyclic groups
cites work: Q4519447
cites work: Direct products of simple modules over Dedekind domains.
cites work: Q4519450
cites work: Purity and algebraic compactness for modules
cites work: Q5732722
DOI: 10.1016/J.JALGEBRA.2008.11.004
description: scientific article; zbMATH DE number 5545847