A Note on Local Weakly SG-Hereditary Domains

  • Kui HuEmail author
  • Jung Wook Lim
  • De Chuan Zhou
Original Paper


A ring R is called weakly SG-hereditary if every ideal of R is SG-projective. In this note, we prove that a local domain R is a Noetherian Warfield domain if and only if it is weakly SG-hereditary. Furthermore, we prove that any countably generated submodule of any free module over a Noetherian local Warfield domain is SG-projective.


Noetherian local Warfield domains SG-Dedekind domains Weakly SG-hereditary rings 

Mathematics Subject Classification

13G05 13D03 



This work was partially supported by the Department of Mathematics of Kyungpook National University and National Natural Science Foundation of China (Grant no. 11671283). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2017R1C1B1008085).


  1. 1.
    Bass, H.: Torsion-free and projective modules. Trans. Am. Math. Soc. 102, 319–27 (1962)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bennis, D., Mahdou, N.: Strongly Gorenstein projective, injective, and flat modules. J. Pure Appl. Algebra 210, 437–445 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bazzoni, S., Salce, L.: Warfield domains. J. Algebra 185, 836–868 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Enochs, E., Jenda, O.: Gorenstein injective and projective modules. Math. Z. 220, 611–633 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fuchs, L., Salce, L.: Modules Over Non-noetherian Domains. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  6. 6.
    Hu, K., Wang, F., Xu, L., Zhao, S.: On overrings of Gorenstein Dedekind domains. J. Korean Math. Soc. 50, 991–1008 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu, K., Kim, H., Wang, F., Xu, L., Zhou, D.: On strongly Gorenstein hereditary rings. Bull. Korean Math. Soc. 56(2), 373–382 (2019)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kaplansky, I.: Commutative Rings. Revised Edition. University of Chicago Press, Chicago (1974)zbMATHGoogle Scholar
  9. 9.
    Matlis, E.: The two-generator problem for ideals. Mich. Math. J. 17, 257–65 (1970)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mahdou, N., Tamekkante, M.: On (strongly) Gorenstein (semi)hereditary rings. Arab. J. Sci. Eng. 36, 431–440 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rotman, J.J.: An Introduction to Homological Algebra, 2nd edn. Springer Science+Business Media, LLC, New York (2009)CrossRefGoogle Scholar
  12. 12.
    Wang, F., Kim, H.: Foundations of Commutative Rings and Their Modules. Springer, Singapore (2016)CrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.College of ScienceSouthwest University of Science and TechnologyMianyangPeople’s Republic of China
  2. 2.Department of MathematicsKyungpook National UniversityDaeguRepublic of Korea

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