Rings whose modules have maximal or minimal injectivity domains.
type: Item
instance of: scholarly article
title: Rings whose modules have maximal or minimal injectivity domains.
zbMATH Open document ID: 1227.16004
author: Noyan Er
author: N. Sokmez
author: Sergio R. López-Permouth
published in: Journal of Algebra
publication date: 2011-07-25T00:00:00Z
review text: Let \(M\) be a unital right \(R\)-module and \(R\) an associative ring with identity. The domain of injectivity of \(M\) is denoted by \(\text{In}^{-1}(M)\), i.e., \(\text{In}^{-1}(M)=\{N\in\text{Mod-}R\mid M\) is \(N\)-injective\}. Of course this domain always includes the semisimple modules; a module has been called poor if \(\text{In}^{-1}(M)\) consists only of semisimple modules [\textit{A. N. Alahmadi, M. Alkan} and \textit{S. R. López-Permouth}, Glasg. Math. J. 52A, 7-17 (2010; Zbl 1228.16004)]. Two constructions of poor modules are given, thereby showing that every ring has a poor module. A module \(M\) is said to crumble if the socles of all factors of \(M\) split. In the paper referred to above, certain semisimple modules that are poor were found; here it is shown that a ring \(R\) has a semisimple poor right module if and only if every (cyclic) right \(R\)-module that crumbles is semisimple, if and only if the only locally Noetherian right \(V\)-modules are the semisimple ones. In the paper of Alahmadi, Alkan and López-Permouth [see reference above], a ring \(R\) was defined to have no (right) middle class if each right \(R\)-module is either injective or poor. Investigating this concept further, it is shown that a ring with no right middle class is the ring direct sum of a semisimple Artinian ring and a ring \(T\) which is either zero or one of the following types: (i) Morita equivalent to a right PCI-domain, (ii) an indecomposable right SI-ring which is either right Artinian or a ring \(V\)-ring, and such that \(\text{soc}(T_T)\) is homogeneous and essential in \(T_T\) and \(T\) has a unique simple singular right module, or (iii) an indecomposable right Artinian ring with homogeneous right socle coinciding with the Jacobson radical and the right singular ideal, and with unique non-injective simple right module. In the third case, \(T\) is either a QF-ring with \(J(T)^2=0\) or poor as a right module. The proof is done by means of a series of lemmas which …
reviewed by: Frieda Theron
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D10
zbMATH DE Number: 5931654
zbMATH Keywords: injective modules
zbMATH Keywords: poor modules
zbMATH Keywords: injectivity domains
zbMATH Keywords: middle classes
zbMATH Keywords: semisimple modules
zbMATH Keywords: direct sums
zbMATH Keywords: V-rings
zbMATH Keywords: PCI-domains
zbMATH Keywords: Noetherian rings
zbMATH Keywords: Artinian rings
zbMATH Keywords: SI-rings
zbMATH Keywords: QF-rings
Wikidata QID: Q123240348
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.10.038
OpenAlex ID: W1963795372
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DOI: 10.1016/J.JALGEBRA.2010.10.038
description: scientific article; zbMATH DE number 5931654