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Subprojectivity in Abelian Categories


In the last few years, López-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as a tool to measure, somehow, the projectivity level of such a module (so not just to determine whether or not the module is projective). In this paper we develop a new treatment of the subprojectivity in any abelian category which shed more light on some of its various important aspects. Namely, in terms of subprojectivity, some classical results are unified and some classical rings are characterized. It is also shown that, in some categories, the subprojectivity measures notions other than the projectivity. Furthermore, this new approach allows, in addition to establishing nice generalizations of known results, to construct various new examples such as the subprojectivity domain of the class of Gorenstein projective objects, the class of semi-projective complexes and particular types of representations of a finite linear quiver. The paper ends with a study showing that the fact that a subprojectivity domain of a class coincides with its first right Ext-orthogonal class can be characterized in terms of the existence of preenvelopes and precovers.

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  1. 1.

    Alagöz, Y., Büyükas, E.: Max-projective modules. J. Algebra Appl. (2021).

  2. 2.

    Bennis, D., Hu, K., Wang, F.: On \(2\)-SG-semisimple rings. Rocky Mountain J. Math. 45, 1093–1100 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bennis, D., Mahdou, N.: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210, 437–445 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bennis, D., Mahbou, N.: Global Gorenstein dimensions. Proc. Amer. Math. Soc. 138, 461–465 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bennis, D., Mahdou, N., Ouarghi, K.: Rings over which all modules are strongly Gorenstein projective. Rocky Mountain J. Math. 40, 749–759 (2010)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Christensen, L.W., Foxby, H.-B.: Hyperhomological Algebra with Applications to Commutative Rings, preprint (2006).

  7. 7.

    Christensen, L.W., Frankild, A., Holm, H.: On Gorenstein projective, injective and flat dimensions-a functorial description with applications. J. Algebra 302, 231–279 (2006)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Durğun, Y.: Rings whose modules have maximal or minimal subprojectivity domain. J. Algebra Appl. 14, 1550083 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Durğun, Y.: On subprojectivity domains of pure-projective modules. J. Algebra Appl. 19, 2050091 (2020)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Enochs, E., Estrada, S.: Projective representations of quivers. Comm. Algebra 33, 3467–3478 (2005)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Enochs, E.E., Jenda, O.M.G.: Relative homological algebra. Walter de Gruyter, Berlin-New York (2000)

    Book  Google Scholar 

  12. 12.

    Fieldhouse, D.J.: Characterizations of modules. Canad. J. Math. 23, 608–610 (1971)

    MathSciNet  Article  Google Scholar 

  13. 13.

    García Rozas, J.R.: Covers and Envelopes in the Category of Complexes of Modules. Chapman & Hall/CRC, Research Notes in Mathematics, 407 (1999)

  14. 14.

    Herzog, I.: Pure-injective envelopes. J. Algebra Appl. 2, 397–402 (2003)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Holston, C., López-Permouth, S.R., Mastromatteo, J., Simental-Rodriguez, J.E.: An alternative perspective on projectivity of modules. Glasgow Math. J. 57, 83–99 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Mao, L.: When does every simple module have a projective envelope? Comm. Algebra 35, 1505–1516 (2007)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Park, S.: Projective representations of quivers. Int. J. Math. & Math. Sci. 31, 97–101 (2002)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Parra, R., Rada, J.: Projective envelopes of finitely generated modules. Algebra Colloq. 18, 801–806 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Popescu, N., Popescu, L.: Theory of categories. Editura Academiei Republicii Socialiste România (1979)

  20. 20.

    Rada, J., Saorín, M.: Rings characterized by (pre)envelopes and (pre)covers of their modules. Comm. Algebra 26, 899–912 (1998)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Stenström, B.: Purity in functor categories. J. Algebra 8, 352–361 (1968)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Trlifaj, J.: Whitehead test modules. Trans. Amer. Math. Soc. 348, 1521–1554 (1996)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Warfield, R.B.: Purity and algebraic compactness for modules. Pacific J. Math. 28, 699–719 (1969)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Zhao, T., Huang, Z.: Special precovered categories of Gorenstein categories. Sci. China Ser. A 62, 1553–1566 (2019)

    MathSciNet  Article  Google Scholar 

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The authors want to express their gratitude to the referee for the very helpful comments and suggestions. Part of this work was done while Amzil and Ouberka were visiting the University of Almería under the Erasmus+ KA107 scholarship grant from November 2019 to July 2020. They would like to express their sincere thanks for the warm hospitality and the excellent working conditions. Also, during the preparation of this paper, Bennis has been invited by the University of Almería. He is always thankful to the department of Mathematics for the warm hospitality during his stay in Almería. The authors Luis Oyonarte and J.R. García Rozas were partially supported by Ministerio de Economía y Competitividad, grant reference 2017MTM2017-86987-P.

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Correspondence to Driss Bennis.

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Amzil, H., Bennis, D., Rozas, J.R.G. et al. Subprojectivity in Abelian Categories. Appl Categor Struct 29, 889–913 (2021).

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  • Subprojectivity
  • Subprojectivity domain
  • Precovers
  • Preenvelopes

Mathematics Subject Classification

  • 18G25