Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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A RECENT GENERALIZATION OF COFINITELY INJECTIVE MODULES

  • Esra OZTURK SOZEN (Faculty of Sciences and Arts, Department of Mathematics, Sinop University)
  • Received : 2021.11.25
  • Accepted : 2023.03.06
  • Published : 2023.09.14
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Abstract

Let R be an associative ring with identity and M be a left R-module. In this paper, we define modules that have the property (δ-CE) ((δ-CEE)), these are modules that have a δ-supplement (ample δ-supplements) in every cofinite extension which are generalized version of modules that have the properties (CE) and (CEE) introduced in [6] and so a generalization of Zöschinger's modules with the properties (E) and (EE) given in [23]. We investigate various properties of these modules along with examples. In particular we prove these: (1) a module M has the property (δ-CEE) if and only if every submodule of M has the property (δ-CE); (2) direct summands of a module that has the property (δ-CE) also have the property (δ-CE); (3) each factor module of a module that has the property (δ-CE) also has the property (δ-CE) under a special condition; (4) every module with composition series has the property (δ-CE); (5) over a δ-V -ring a module M has the property (δ-CE) if and only if M is cofinitely injective; (6) a ring R is δ-semiperfect if and only if every left R-module has the property (δ-CE).

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References

  1. Khaled Al-Takhman, Cofinitely δ-supplemented and Cofinitely δ-semiperfect modules, Int. J. Algebra 1 (2007), no. 12, 601-613.
  2. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer Science & Business Media, 2012.
  3. P. Aydogdu, Rings over which every module has a flat δ-cover, Turkish J. Math. 37 (2013), no. 1, 182-194.
  4. H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), no. 3, 466-488.
  5. S. Bazzoni and L. Salce, Strongly flat covers, Journal of the London Mathematical Society 66 (2002), no. 2, 276-294.
  6. H. Calisici and E. Turkmen, Modules that have a supplement in every cofinite extension, Georgian Math. J. 19 (2012), no. 2, 209-216.
  7. K. Goodearl, Ring theory: Nonsingular Rings and Modules, CRC Press, 1976.
  8. F. Kasch, Locally injective modules and locally projective modules, Rocky Mountain J. Math. 32 (2002), no. 4, 1493-1504.
  9. F. R. Kasch and E. A. Mares, Eine kennzeichnung semi-perfekter moduln, Nagoya Math. 27 (1966), no. 2, 525-529.
  10. M. T. Kosan, δ-lifting and δ-supplemented modules, Algebra Colloq. 14 (2007), no. 1, 53-60.
  11. E. Onal, H. Calisici, and E. Turkmen, Modules that have a weak supplement in every extension, Miskolc Math. Notes 17 (2016), no. 1, 471-481.
  12. S. Ozdemir, Rad-supplementing modules, J. Korean Math. Soc. 53 (2016), no. 2, 403-414.
  13. E. Sozen and S Eren, Modules that Have a δ-supplement in every extension, Eur. J. of Pure Appl. Math. 10 (2017), no. 4, 730-738.
  14. E. Sozen, F. Eryilmaz, and S Eren, Modules that have a weak supplement in every torsion extension, Journal of Science and Arts 39 (2017), no. 2, 269-274.
  15. Y. Talebi and B. Talaee, On generalized δ-supplemented modules, Vietnam J. Math. 37 (2009), no. 4, 515-525.
  16. R. Tribak, When finitely generated δ-supplemented modules are supplemented, Algebra Colloq. 22 (2015), no. 1, 119-130.
  17. B. N. Turkmen, Modules that have a supplement in every coatomic extension, Miskolc Math. Notes 16 (2015), no. 1, 543-551.
  18. B. N. Turkmen, Modules that have a rad-supplement in every cofinite extension, Miskolc Math. Notes 14 (2013), no. 3, 1059-1066.
  19. B. Ungor, S. Halicioglu, and A. Harmanci, On a class of δ-supplemented modules, Bull. Malays. Math. Sci. Soc. 37 (2014), no. 3, 703-717.
  20. R. Wisbauer, Foundations of Module and Ring Theory, Routledge, 2018.
  21. Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7 (2000), no. 3, 305-318.
  22. B. Zimmermann-Huisgen, Pure submodules of direct products of free modules, Math. Ann. 224 (1974), no. 3, 233-245.
  23. H. Zoschinger, Moduln die in jeder erweiterung ein komplement haben, Math. Scand. 35 (1975), no. 2, 267-287.