Dv Huynh Sk Jain on Weakly Injective Continuous Modules

Abstract

We consider two types of weak-injectivity domains of a module. Small weak-injectivity domains gauge the extent of weak injectivity of modules; weak-injectivity domains are an alternative way to gauge their injectivity. We focus on notions which are opposite, respectively, to weakly injective or injective modules according to these two schemes. While we do consider both notions to some extent, our emphasis is on modules with smallest weak-injectivity domain; we name those modules extremely poor. The intersection of all (small) domains of weak injectivity is the natural focus of attention. The intersection of all (small) domains of weak injectivity contains the class of all (finitely generated) semisimple modules, but it is not always equal to it. The domain of weak injectivity of an extremely poor module thusly depends on the ring in question. We consider necessary and sufficient conditions for said intersections to consist solely of semisimple modules and look at examples of both circumstances, when this is the case and when it is not. When the intersection of weak-injectivity domains consists of semisimple modules, every extremely poor module is poor; we study further connections between these two notions. Comparisons are also drawn to paupers (poor modules with no proper poor direct summands); paupers are a type of poor module that is significant in understanding the intrinsic structure of poor modules. Extremely poor modules over PCI domains and extremely poor abelian groups are given particular attention.

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Acknowledgements

This work was done during the visit of the first and third authors to the Ohio University, Center of Ring Theory and its Applications. They would like to thank all the members of the Center for their warm hospitality. The first author also acknowledges the financial support she received from Hacettepe University, Scientific Research Projects Coordination Unit, Project No: FBI-2017-14952 and TUBITAK Project No: 117F084. The third author wants to thank the Fulbright-García Robles Program and COMEXUS for the scholarship that allowed her to carry out this postdoctoral stay.

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Authors and Affiliations

1. Department of Mathematics, Hacettepe University, 06800, Beytepe Ankara, Turkey

   Pınar Aydoğdu

2. Center of Ring Theory and its Applications, Ohio University, Athens, OH, 45701, USA

   Pınar Aydoğdu & Martha Lizbeth Shaid Sandoval-Miranda

3. Department of Mathematics, Ohio University, Athens, OH, 45701, USA

   Sergio R. López-Permouth

4. Department of Mathematics, Metropolitan Autonomous University, Mexico City, Mexico

   Martha Lizbeth Shaid Sandoval-Miranda

Corresponding author

Correspondence to Pınar Aydoğdu.

Additional information

Communicated by See Keong Lee.

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Aydoğdu, P., López-Permouth, S.R. & Sandoval-Miranda, M.L.S. On the Weak-Injectivity Profile of a Ring. Bull. Malays. Math. Sci. Soc. 44, 35–53 (2021). https://doi.org/10.1007/s40840-020-00938-3

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• Received: 02 September 2019

• Revised: 30 March 2020

• Published: 27 April 2020

• Issue Date: January 2021

• DOI : https://doi.org/10.1007/s40840-020-00938-3

Keywords

• Weakly-injective modules
• (Extremely)poor modules
• Weak-injectivity domains

Mathematics Subject Classification

• 16D10
• 16D50
• 16D99

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