Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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FLAT DIMENSIONS OF INJECTIVE MODULES OVER DOMAINS

  • Hu, Kui (College of Science Southwest University of Science and Technology) ;
  • Lim, Jung Wook (Department of Mathematics Kyungpook National University) ;
  • Zhou, De Chuan (College of Science Southwest University of Science and Technology)
  • Received : 2019.08.13
  • Accepted : 2019.11.06
  • Published : 2020.07.31
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Abstract

Let R be a domain. It is proved that R is coherent when IFD(R) ⩽ 1, and R is Noetherian when IPD(R) ⩽ 1. Consequently, R is a G-Prüfer domain if and only if IFD(R) ⩽ 1, if and only if wG-gldim(R) ⩽ 1; and R is a G-Dedekind domain if and only if IPD(R) ⩽ 1.

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References

  1. D. Bennis, Weak Gorenstein global dimension, Int. Electron. J. Algebra 8 (2010), 140-152.
  2. 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
  3. N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math. 78 (1993), no. 2, 165-177. https://doi.org/10.1007/BF02599307
  4. E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, flat resolvents and dimensions, Comment. Math. Univ. Carolin. 34 (1993), no. 2, 203-211.
  5. X. Fu, H. Zhu, and N. Ding, On copure projective modules and copure projective dimensions, Comm. Algebra 40 (2012), no. 1, 343-359. https://doi.org/10.1080/00927872.2010.531337
  6. H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1279-1283. https://doi.org/10.1090/S0002-9939-03-07466-5
  7. 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
  8. L. Qiao and F. Wang, A Gorenstein analogue of a result of Bertin, J. Algebra Appl. 14 (2015), no. 2, 1550019, 13 pp. https://doi.org/10.1142/S021949881550019X
  9. 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
  10. S. Xing and H. Kim, Generalized coherent domains of self-weak injective dimension flat most one, Comm. Algebra 47 (2019), no. 5, 1908-1916. https://doi.org/10.1080/00927872.2018.1524006