Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A NOTE ON ARTINIAN LOCAL RINGS

  • Hu, Kui (College of Science Southwest University of Science and Technology) ;
  • Kim, Hwankoo (Division of Computer Engineering Hoseo University) ;
  • Zhou, Dechuan (College of Science Southwest University of Science and Technology)
  • Received : 2021.10.17
  • Accepted : 2022.02.23
  • Published : 2022.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this note, we prove that an Artinian local ring is G-semisimple (resp., SG-semisimple, 2-SG-semisimple) if and only if its maximal ideal is G-projective (resp., SG-projective, 2-SG-projective). As a corollary, we obtain the global statement of the above. We also give some examples of local G-semisimple rings whose maximal ideals are n-generated for some positive integer n.

Keywords

Acknowledgement

The authors would like to express their sincere thanks for the reviewer for his/her careful reading and an interesting suggestion.

References

  1. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, AddisonWesley Publishing Co., Reading, MA, 1969.
  2. D. Bennis, (n, m)-SG rings, Arab. J. Sci. Eng. ASJE. Math. 35 (2010), no. 2D, 169-178.
  3. D. Bennis and F. Couchot, On 1-semiregular and 2-semiregular rings, J. Algebra Appl. 20 (2021), no. 12, Paper No. 2150221, 21 pp. https://doi.org/10.1142/S0219498821502212
  4. D. Bennis, K. Hu, and F. Wang, On 2-SG-semisimple rings, Rocky Mountain J. Math. 45 (2015), no. 4, 1093-1100. https://doi.org/10.1216/RMJ-2015-45-4-1093
  5. D. Bennis and N. Mahdou, Strongly Gorenstein projective, injective, and flat modules, J. Pure Appl. Algebra 210 (2007), no. 2, 437-445. https://doi.org/10.1016/j.jpaa.2006.10.010
  6. D. Bennis and N. Mahdou, A generalization of strongly Gorenstein projective modules, J. Algebra Appl. 8 (2009), no. 2, 219-227. https://doi.org/10.1142/S021949880900328X
  7. D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138 (2010), no. 2, 461-465. https://doi.org/10.1090/S0002-9939-09-10099-0
  8. D. Bennis, N. Mahdou, and K. Ouarghi, Rings over which all modules are strongly Gorenstein projective, Rocky Mountain J. Math. 40 (2010), no. 3, 749-759. https://doi.org/10.1216/RMJ-2010-40-3-749
  9. E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611-633. https://doi.org/10.1007/BF02572634
  10. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter Co., Berlin, 2000.
  11. H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 167-193. https://doi.org/10.1016/j.jpaa.2003.11.007
  12. K. Hu, J. W. Lim, and D. C. Zhou, Some change of rings results on (n, m)-SG modules and (n, m)-SG rings, Comm. Algebra 48 (2020), no. 9, 4037-4050. https://doi.org/10.1080/00927872.2020.1754843
  13. K. Hu and F. Wang, Some results on Gorenstein Dedekind domains and their factor rings, Comm. Algebra 41 (2013), no. 1, 284-293. https://doi.org/10.1080/00927872.2011.629268
  14. K. Hu, F. Wang, L. Xu, and S. Zhao, On overrings of Gorenstein Dedekind domains, J. Korean Math. Soc. 50 (2013), no. 5, 991-1008. https://doi.org/10.4134/JKMS.2013.50.5.991
  15. I. Kaplansky, Commutative Rings, revised edition, The University of Chicago Press, Chicago, IL, 1974.
  16. K. R. McLean, Commutative Artinian principal ideal rings, Proc. London Math. Soc. (3) 26 (1973), 249-272. https://doi.org/10.1112/plms/s3-26.2.249
  17. J. J. Rotman, An Introduction to Homological Algebra, second edition, Universitext, Springer, New York, 2009. https://doi.org/10.1007/b98977
  18. The Stacks Project Authors, Stacks Project, 2022. https://stacks.math.columbia.edu/tag/00J4
  19. F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016. https://doi.org/10.1007/978-981-10-3337-7
  20. G. Zhao and Z. Huang, n-strongly Gorenstein projective, injective and flat modules, Comm. Algebra 39 (2011), no. 8, 3044-3062. https://doi.org/10.1080/00927872.2010.496749