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dc.contributor.authorAlizade, Rafail
dc.contributor.authorBüyükaşık, Engin
dc.contributor.authorLópez-Permouth, Sergio
dc.contributor.authorYang, Liu
dc.date.accessioned2020-01-14T11:20:13Z
dc.date.available2020-01-14T11:20:13Z
dc.date.issued2018-05en_US
dc.identifier.citationAlizade, R., Büyükaşık, E., López-Permouth, S., and Yang, L. (2018). Poor modules with no proper poor direct summands. Journal of Algebra, 502, 24-44. doi:10.1016/j.jalgebra.2017.12.034en_US
dc.identifier.issn0021-8693
dc.identifier.urihttps://doi.org/10.1016/j.jalgebra.2017.12.034
dc.identifier.urihttps://hdl.handle.net/11147/7578
dc.description.abstractAs a mean to provide intrinsic characterizations of poor modules, the notion of a pauper module is introduced. A module is a pauper if it is poor and has no proper poor direct summand. We show that not all rings have pauper modules and explore conditions for their existence. In addition, we ponder the role of paupers in the characterization of poor modules over those rings that do have them by considering two possible types of ubiquity: one according to which every poor module contains a pauper direct summand and a second one according to which every poor module contains a pauper as a pure submodule. The second condition holds for the ring of integers and is just as significant as the first one for Noetherian rings since, in that context, modules having poor pure submodules must themselves be poor. It is shown that the existence of paupers is equivalent to the Noetherian condition for rings with no middle class. As indecomposable poor modules are pauper, we study rings with no indecomposable right middle class (i.e. the ring whose indecomposable right modules are pauper or injective). We show that semiartinian V-rings satisfy this property and also that 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. 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.en_US
dc.language.isoengen_US
dc.publisherElsevieren_US
dc.relation.isversionof10.1016/j.jalgebra.2017.12.034en_US
dc.rightsinfo:eu-repo/semantics/openAccessen_US
dc.subjectInjective moduleen_US
dc.subjectPauper moduleen_US
dc.subjectPoor moduleen_US
dc.subjectModules (Algebra)en_US
dc.titlePoor modules with no proper poor direct summandsen_US
dc.typearticleen_US
dc.contributor.authorID0000-0003-2402-3496en_US
dc.contributor.iztechauthorBüyükaşık, Engin
dc.relation.journalJournal of Algebraen_US
dc.contributor.departmentIzmir Institute of Technology. Mathematicsen_US
dc.identifier.volume502en_US
dc.identifier.startpage24en_US
dc.identifier.endpage44en_US
dc.identifier.wosWOS:000428834300002
dc.identifier.scopusSCOPUS:2-s2.0-85044329219
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanıen_US


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