On \(\varvec{\mathcal {D}}\)-closed submodules

Proceedings - Mathematical Sciences volume 130, Article number: 1 (2020) Cite this article

Abstract

A submodule N of a module M is called \(\mathcal {D}\)-closed if the socle of M / N is zero. \(\mathcal {D}\)-closed submodules are similar to \(\mathcal {S}\)-closed submodules (a generalization of closed submodules) defined through nonsingular modules. First, we describe the smallest proper class (due to Buchsbaum) containing the class of short exact sequences determined by \(\mathcal {D}\)-closed submodules in terms of that submodule, and show that it coincides with other classes of modules under certain conditions. Second, we study coprojective modules of this class, called edc-flat modules. We give some equivalent conditions for injective modules to be edc-flat for special rings, and for edc-flat modules to be projective (flat) for any ring.

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Acknowledgements

The authors would like to thank the referee for the review of this manuscript.

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Affiliations

  1. Department of Mathematics, Çukurova University, Adana, Turkey

    Yılmaz Durǧun

  2. Department of Mathematics, Dokuz Eylul University, Izmir, Turkey

    Salahattin Özdemir

Authors
  1. Yılmaz Durǧun
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  2. Salahattin Özdemir
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Correspondence to Salahattin Özdemir.

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Communicating Editor: B Sury

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Cite this article

Durǧun, Y., Özdemir, S. On \(\varvec{\mathcal {D}}\)-closed submodules. Proc Math Sci 130, 1 (2020). https://doi.org/10.1007/s12044-019-0537-1

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Keywords

  • Closed submodule
  • flat module
  • Dickson torsion theory
  • semiartinian module
  • proper class

Mathematics Subject Classification

  • 16D40
  • 18G25