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


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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  1. 1.

    Buchsbaum D A, A note on homology in categories, Ann. Math.69(1) (1959) 66–74

    MathSciNet  Article  Google Scholar 

  2. 2.

    Büyükaşık E and Durğun Y, Absolutely \(s\)-pure modules and neat-flat modules, Comm. Algebra43(2) (2015) 384–399

    MathSciNet  Article  Google Scholar 

  3. 3.

    Büyükaşık E and Durğun Y, Neat-flat modules, Comm. Algebra44(1) (2016) 416–428

    MathSciNet  Article  Google Scholar 

  4. 4.

    Clark J, Lomp C, Vanaja N and Wisbauer R, Lifting Modules (2006) (Basel: Birkhäuser Verlag)

  5. 5.

    Cohn P M, On the free product of associative rings, Math. Z.71 (1959) 380–398

    MathSciNet  Article  Google Scholar 

  6. 6.

    Crivei S and Keskin Tütüncü D, Relatively divisible and relatively flat objects in exact categories, https://arxiv.org/pdf/1810.11637.pdf

  7. 7.

    Crivei S and Şahinkaya S, Modules whose closed submodules with essential socle are direct summands, Taiwanese J. Math.18(4) (2014) 989–1002

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dung N V, Huynh D V, Smith P F and Wisbauer R, Extending Modules, Pitman Research Notes in Mathematical Series 313 (1994) (Harlow: Longman Scientific & Technical)

  9. 9.

    Durğun Y and Özdemir S, On \(\cal{S}\)-closed submodules, J. Korean Math. Soc.54(4) (2017) 1281–1299

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Fuchs L, Neat submodules over integral domains, Period. Math. Hungar.64(2) (2012) 131–143

    MathSciNet  Article  Google Scholar 

  11. 11.

    Generalov A I, On weak and \(w\)-high purity in the category of modules, Math. USSR, Sb.34 (1978) 345–356, translated from Russian from Mat. Sb., N. Ser.105(147) 389–402

  12. 12.

    Goodearl K R, Singular torsion and the splitting properties, 124 (1972) (Providence, RI: American Mathematical Society)

  13. 13.

    Goodearl K R, Ring Theory (1976) (New York-Basel: Marcel Dekker Inc.)

  14. 14.

    Honda K, Realism in the theory of abelian groups I, Comment. Math. Univ. St. Paul.5 (1956) 37–75

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Kara Y and Tercan A, When some complement of a \(z\)-closed submodule is a summand, Comm. Algebra46(7) (2018) 3071–3078

    MathSciNet  Article  Google Scholar 

  16. 16.

    Keller B, Chain complexes and stable categories, Manuscripta Math.67 (1990) 379–417

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kepka T, On one class of purities, Comment. Math. Univ. Carolinae14 (1973) 139–154

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Lam T Y, Lectures on modules and rings, Graduate Texts in Mathematics (1999) (New York: Springer-Verlag)

  19. 19.

    Mišina A P and Skornjakov L A, Abelian Groups and Modules, Algebra, Logic and Applications, AMS Translational Series: 2, 107 (1976) (Providence, RI)

  20. 20.

    Nicholson W K and Yousif M F, Quasi-Frobenius rings, Cambridge Tracts in Mathematics 158 (2003) (Cambridge University Press)

  21. 21.

    Quillen D, Higher algebraic K-theory: I, Lecture Notes in Mathematics, vol. 341 (1973) (Berlin: Springer) pp. 85–147

  22. 22.

    Renault G, Étude de certains anneaux liés aux sous-modules compléments d’un \(A\)-module, C. R. Acad. Sci. Paris259 (1964) 4203–4205

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Rotman J J, An introduction to homological algebra, 2nd edition, Universitext (2009) (New York: Springer)

  24. 24.

    Sklyarenko E G, Relative homological algebra in the category of modules, Usp. Mat. Nauk.33(3) (1978) 85–120

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Stenström B T, High submodules and purity, Ark. Mat.7 (1967) 173–176

    MathSciNet  Article  Google Scholar 

  26. 26.

    Zöschinger H, Schwach-Flache Moduln, Comm. Algebra41(12) (2013) 4393–4407

    MathSciNet  Article  Google Scholar 

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

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Correspondence to Salahattin Özdemir.

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Communicating Editor: B Sury

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

Mathematics Subject Classification

  • 16D40
  • 18G25