Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GOLDIE EXTENDING PROPERTY ON THE CLASS OF z-CLOSED SUBMODULES

  • Tercan, Adnan (Department of Mathematics Hacettepe University) ;
  • Yasar, Ramazan (Hacettepe-ASO 1.OSB Vocational School Hacettepe University) ;
  • Yucel, Canan Celep (Pamukkale University Department of Mathematics Faculty of Arts and Sciences)
  • Received : 2021.04.29
  • Accepted : 2021.08.27
  • Published : 2022.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this article, we define a module M to be Gz-extending if and only if for each z-closed submodule X of M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We investigate structural properties of Gz-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for Gz-extending modules. We obtain that if a ring is right Gz-extending, then so is its essential overring. Also it is shown that the Gz-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right Gz-extending ring.

Keywords

Acknowledgement

The authors would like to express their appreciation to the referee for her/his careful reading of the paper as well as valuable suggestions.

References

  1. E. Akalan, G. F. Birkenmeier, and A. Tercan, Goldie extending modules, Comm. Algebra 37 (2009), no. 2, 663-683. https://doi.org/10.1080/00927870802254843
  2. E. Akalan, G. F. Birkenmeier, and A. Tercan, Goldie extending rings, Comm. Algebra 40 (2012), no. 2, 423-428. https://doi.org/10.1080/00927872.2010.529096
  3. G. F. Birkenmeier, J. K. Park, and S. T. Rizvi, Ring hulls and applications, J. Algebra 304 (2006), no. 2, 633-665. https://doi.org/10.1016/j.jalgebra.2006.06.034
  4. G. F. Birkenmeier, J. K. Park, and S. T. Rizvi, Extensions of Rings and Modules, Birkhauser/Springer, New York, 2013. https://doi.org/10.1007/978-0-387-92716-9
  5. V. Camillo, I. Herzog, and P. P. Nielsen, Non-self-injective injective hulls with compatible multiplication, J. Algebra 314 (2007), no. 1, 471-478. https://doi.org/10.1016/j.jalgebra.2007.03.016
  6. S. Crivei and S. S,ahinkaya, Modules whose closed submodules with essential socle are direct summands, Taiwanese J. Math. 18 (2014), no. 4, 989-1002. https://doi.org/10.11650/tjm.18.2014.3388
  7. N. V. Dung, D. Van Huynh, P. F. Smith, and R. Wisbauer, Extending modules, Pitman Research Notes in Mathematics Series v.313, Longman, Harlow, 1994.
  8. L. Fuchs, Infinite Abelian Groups, Academic press, 1970.
  9. A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201-220. https://doi.org/10.1112/plms/s3-10.1.201
  10. K. R. Goodearl, Ring Theory, Pure and Applied Mathematics, No. 33, Marcel Dekker, Inc., New York, 1976.
  11. Y. Kara and A. Tercan, When some complement of a z-closed submodule is a summand, Comm. Algebra 46 (2018), no. 7, 3071-3078. https://doi.org/10.1080/00927872.2017.1404080
  12. I. Kikumasa and Y. Kuratomi, On Goldie extending modules with condition (C2), Comm. Algebra 44 (2016), no. 9, 4041-4046. https://doi.org/10.1080/00927872.2015.1087536
  13. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4612-0525-8
  14. S. H. Mohamed and B. J. Muller, Ojective modules, Comm. Algebra 30 (2002), no. 4, 1817-1827. https://doi.org/10.1081/AGB-120013218
  15. B. L. Osofsky, Homological properties of rings and modules, Ph.D thesis, Rutgers University, New Jersey, 1964.
  16. B. L. Osofsky, On ring properties of injective hulls, Canad. Math. Bull. 10 (1967), 275-282. https://doi.org/10.4153/CMB-1967-027-9
  17. D. S. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley-Interscience, New York, 1977.
  18. P. F. Smith, Modules for which every submodule has a unique closure, in Ring theory (Granville, OH, 1992), 302-313, World Sci. Publ., River Edge, NJ, 1993.
  19. P. F. Smith and A. Tercan, Continuous and quasi-continuous modules, Houston J. Math. 18 (1992), no. 3, 339-348.
  20. B. Stenstrom, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York, 1975.
  21. A. Tercan, On certain CS-rings, Comm. Algebra 23 (1995), no. 2, 405-419. https://doi.org/10.1080/00927879508825228
  22. A. Tercan, On CLS-modules, Rocky Mountain J. Math. 25 (1995), no. 4, 1557-1564. https://doi.org/10.1216/rmjm/1181072161
  23. A. Tercan and C. C. Yucel, Module theory, extending modules and generalizations, Frontiers in Mathematics, Birkhauser/Springer, 2016. https://doi.org/10.1007/978-3-0348-0952-8