Rings whose cyclic modules have restricted injectivity domains

Er, Noyan; López-Permouth, Sergio R.; Tung, Nguyen Khanh

doi:10.1016/j.jalgebra.2016.08.003

Cited by 12 publications

(4 citation statements)

References 8 publications

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“…Observe that R is a ring with no middle class that is not semisimple Artinian if and only if SSMod-R is the unique coatom in the lattice iP(R). Rings with no middle class have been introduced in [1] and have been extensively studied in several papers (see, for example, [9,10,11,15,18]) from which we see that despite its quite simple profile, the structure of a ring with no middle class that is not semisimple Artinian can be too complicated. A key property for addressing challenges in the study of the structure of a ring R with no middle class is the existence of a ring decomposition R = S × T , where S is a semisimple Artinian ring and T is either zero or an indecomposable ring with no middle class such that the right socle of T is either zero or essential in T T .…”

Section: Introductionmentioning

confidence: 99%

On the Splitting Property of Rings with Restricted Class of Injectivity Domains

Saraç

2023

Preprint

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There has been a growing research interest in exploring rings whose modules are either injective or far from being injective to a specified degree. As is commonly observed in each of such scenarios, rings can be broken as a direct sum, where one of the two components is a semisimple Artinian ring and the other is an indecomposable ring having zero or (essential) homogeneous right socle. Although this phenomenon has become increasingly expected and less surprising with every new study, proving it has never become easier, and in most cases, it has been obtained as a result of considerable efforts. This work aims to bring such studies to a common denominator and tries to explain why this phenomenon occurs each time we deal with the same type of problem. Mathematics Subject Classification. Primary 16D50; Secondary 16S90, 16N60

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Section: Introductionmentioning

confidence: 99%

On the Splitting Property of Rings with Restricted Class of Injectivity Domains

Saraç

2023

Preprint

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“…The study of poor modules and rings they serve to characterize has experienced growing interest in recent years (see [8,11,12,19]). Searching for an alternative notion which is also an opposite of injectivity, the notions of subinjectivity and of subinjectivity domain were introduced in [8] and they served to introduce the so-called indigent modules.…”

Section: Introductionmentioning

confidence: 99%

On the structure of modules defined by subinjectivity

2019

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The aim of this paper is to present new results and generalize some results about indigent modules. The commutative rings whose simple modules are indigent or injective are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. We show that a reduced module is indigent if and only if it is a Whitehead test module for injectivity over commutative hereditary noetherian rings. Furthermore, Dedekind domains are characterized by test modules for injectivity by subinjectivity.

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“…Following [1], when no module in A is poor, we say that the ring is an A-utopia and, in the extreme opposite end of the spectrum, when every module in A is poor we say that the ring is A-destitute. In [10], the authors study cyclic-destitute rings without a middle class.…”

Section: Introductionmentioning

confidence: 99%

Poor modules with no proper poor direct summands

et al. 2018

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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. Structure theorems are also provided for rings without indecomposable middle class when the rings are Artinian serial or right Artinian. Rings for which not having an indecomposable middle class suffices not to have a middle class include commutative Noetherian and Artinian serial rings. The structure of poor modules is completely determined over commutative hereditary Noetherian rings. Pauper Abelian groups with torsion-free rank one are fully characterized.

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Rings whose cyclic modules have restricted injectivity domains

Cited by 12 publications

References 8 publications

On the Splitting Property of Rings with Restricted Class of Injectivity Domains

On the Splitting Property of Rings with Restricted Class of Injectivity Domains

On the structure of modules defined by subinjectivity

Poor modules with no proper poor direct summands

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