Poor modules with no proper poor direct summands
type: Item
instance of: scholarly article
title: Poor modules with no proper poor direct summands
zbMATH Open document ID: 1397.16003
author: Peng Zhang
published in: Journal of Algebra
publication date: 2018-03-21T00:00:00Z
full work available at URL: https://hdl.handle.net/11147/7578
review text: \textit{Poor modules} have been introduced by \textit{A. N. Alahmadi} et al. [Glasg. Math. J. 52A, 7--17 (2010; Zbl 1228.16004)], as follows. Let \(R\) be an associative ring with nonzero identity element, and let \(M\) be a unital right \(R\)-module. The module \(M\) is called \textit{poor} if whenever \(M\) is \(N\)-injective for some right \(R\)-module then \(N\) is a semisimple module. In the paper under review, the authors investigate poor modules that have an additional property, they call \textit{pauper}: a module \(M\) is said to be \textit{pauper} if it is poor and has no proper poor direct summand. The ring \(R\) was said to have no right middle class in the paper of Alahmadi et al. [loc. cit.] mentioned above if each right \(R\)-module is either injective or poor. Among others, the authors prove in Sections 3 and 4 the following two results. (1) For a ring \(R\) with no right middle class there exists a pauper right \(R\)-module if and if \(R\) is right Noetherian. (2) A commutative Noetherian ring \(R\) has no indecomposable middle class if and only if \(R\) is the direct product of finitely many fields and at most one ring of composition length 2. Section 4 is devoted to the study of poor and pauper modules over Noetherian rings; for commutative hereditary Noetherian rings the poor modules are fully characterized. Section 5 investigates poor and pauper modules over semi-Artinian rings, while in the last Section 6, abelian groups of torsion-free rank 1 are described.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 16D99
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E99
zbMATH DE Number: 6851766
zbMATH Keywords: injective module
zbMATH Keywords: poor module
zbMATH Keywords: pauper module
zbMATH Keywords: Noetherian ring
reviewed by: Toma Albu
MaRDI profile type: Publication
OpenAlex ID: W2791532510
cites work: POOR MODULES: THE OPPOSITE OF INJECTIVITY
cites work: Poor and pi-poor Abelian groups
cites work: Proper classes related with complements and supplements
cites work: Modules and abelian groups with minimal (pure-) projectivity domains
cites work: Q4023148
cites work: On Artinian rings with restricted class of injectivity domains.
cites work: Rugged modules: The opposite of flatness
cites work: On semi-artinian \(V\)-modules
cites work: Rings whose modules have maximal or minimal injectivity domains.
cites work: Rings whose cyclic modules have restricted injectivity domains
cites work: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules
cites work: Q5673086
cites work: Q4513600
cites work: Singular torsion and the splitting properties
cites work: Q4224184
cites work: Characterizing rings in terms of the extent of the injectivity and projectivity of their modules.
cites work: Modules With Descending Chain Condition
cites work: On rings whose simple modules are injective
cites work: Q3995216
cites work: An introduction to homological algebra
cites work: Q5331572
cites work: Endomorphism Rings of Projective Modules
cites work: Q3267446
DOI: 10.1016/J.JALGEBRA.2017.12.034
Recommended article: Rings whose modules have maximal or minimal injectivity domains.
Recommended article: On Artinian rings with restricted class of injectivity domains.
Recommended article: POOR MODULES: THE OPPOSITE OF INJECTIVITY
Recommended article: Rings whose cyclic modules have restricted injectivity domains
Recommended article: Modules and abelian groups with a bounded domain of injectivity
Recommended article: Modules whose injectivity domains are restricted to semi-artinian modules
Recommended article: Modules and abelian groups with minimal (pure-) projectivity domains
Recommended article: On the weak-injectivity profile of a ring
Recommended article: Subprojectivity domains of pure-projective modules
Recommended article: Rings without a middle class from a lattice-theoretic perspective
description: scientific article; zbMATH DE number 6851766