Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RINGS OF COPURE PROJECTIVE DIMENSION ONE

  • Xiong, Tao (Department of Mathematics Sichuan Normal University)
  • Received : 2016.01.08
  • Published : 2017.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, in terms of the notions of strongly copure projective modules and the copure projective dimension cpD(R) of a ring R were defined in [12], we show that a domain R has $cpD(R){\leq}1$ if and only if R is a Gorenstein Dedekind domain.

Keywords

References

  1. J. Abuhlail and M. Jarrar, Tilting modules over almost perfect domains, J. Pure Appl. Algebra 215 (2011), no. 8, 2024-2033. https://doi.org/10.1016/j.jpaa.2010.11.013
  2. H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. https://doi.org/10.1090/S0002-9947-1960-0157984-8
  3. D. Bennis, A short survey on Gorenstein global dimension, Actes des rencontres du C.I.R.M. 2 (2010), no. 2, 115-117. https://doi.org/10.5802/acirm.46
  4. R. R. Colby, Rings which have flat injective modules, J. Algebra 35 (1975), 239-252. https://doi.org/10.1016/0021-8693(75)90049-6
  5. T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc. 81 (1981), no. 2, 175-177. https://doi.org/10.1090/S0002-9939-1981-0593450-2
  6. N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Mathematica 78 (1993), no. 2, 165-177. https://doi.org/10.1007/BF02599307
  7. N. Q. Ding and J. L. Chen, On copure flat modules and flat resolvents, Comm. Algebra 24 (1996), no. 3, 1071-1081. https://doi.org/10.1080/00927879608825623
  8. 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.
  9. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York 2000.
  10. D. J. Fieldhouse, Character modules, dimension and purity, GlasgowMath. J. 13 (1972), 144-146.
  11. X. H. Fu and N. Q. Ding, On strongly copure flat modules and copure flat dimensions, Comm. Algebra 38 (2010), no. 12, 4531-4544. https://doi.org/10.1080/00927870903428262
  12. X. H. Fu and H. Y. Zhu, and N. Q. 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
  13. Z. H. Gao, n-copure projective modules, Mathematical Notes 97 (2015), no. 1, 50-56. https://doi.org/10.1134/S000143461501006X
  14. R. Gobel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, Berlin-New York: Walter de Gruyter, 2006.
  15. K. Hu and F. G. 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
  16. S. Jain, Flat and FP-injective, Proc. AMS. 41 (1979), 437-442.
  17. C. U. Jensen, On the vanishing of ${\lim_{\leftarrow}^{(i)}}$, J. Algebra 15 (1970), 155-166.
  18. M. Raynaud and L. Gruson, Griteres de plattitude et de projectivite. Techniques de "platification" d'un module, Invent. Math. 13 (1971), 1-89. https://doi.org/10.1007/BF01390094
  19. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York 1979.
  20. J. J. Rotman, An Introduction to Homological Algebra, 2nd ed. New York: Springer Science+Business Media, LLC, 2009.
  21. L. Salce, Almost perfect domains and their modules, In Commutative algebra: Noetherian and non-Noetherian perspectives, pp. 363-386, New York, Springer, 2011.
  22. B. Stenstrom, Coherent rings and FP-injective modules, J. London Math. Soc. 2 (1970), 323-329.
  23. T. Xiong, F. G. Wang, and K. Hu, Copure Projective Modules and CPH-rings, J. Sichuan Normal University (Natural Science) 36 (2013), no. 2, 198-201.