Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

∏-COHERENT DIMENSIONS AND ∏-COHERENT RINGS

  • Mao, Lixin (DEPARTMENT OF BASIC COURSES NANJING INSTITUTE OF TECHNOLOGY)
  • Published : 2007.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

R is called a right ${\Pi}-coherent$ ring in case every finitely generated torsion less right R-module is finitely presented. In this paper, we define a dimension for rings, called ${\Pi}-coherent$ dimension, which measures how far away a ring is from being ${\Pi}-coherent$. This dimension has nice properties when the ring in question is coherent. In addition, we study some properties of ${\Pi}-coherent$ rings in terms of preenvelopes and precovers.

Keywords

References

  1. L. Bican, R. EI Bashir, and E. E. Enochs, All modules have fiat covers, Bull. London Math. Soc. 33 (2001), 385-390 https://doi.org/10.1017/S0024609301008104
  2. L. Bonami, On the Structure of Skew Group Rings, Algebra Berichte 48, Verlag Reinhard Fisher, Munchen, 1984
  3. V. Camillo, Coherence for polynomial rings, J. Algebra 132 (1990), 72-76 https://doi.org/10.1016/0021-8693(90)90252-J
  4. F. C. Chen, J. Y. Tang, Z. Y. Huang, and M. Y. Wang, ${\Pi}$-coherent rings and FGTinjective dimension, Southeast Asian Bulletin Math. 19 (1995), no. 3, 105-112
  5. N. Q. Ding, On envelopes with the unique mapping property, Comm. Algebra 24 (1996), no. 4, 1459-1470 https://doi.org/10.1080/00927879608825646
  6. N. Q. Ding and J. L. Chen, Relative coherence and preenvelopes, Manuscripta Math; 81 (1993), 243-262 https://doi.org/10.1007/BF02567857
  7. P. C. Eklof and J. Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001), no. 1,41-51 https://doi.org/10.1112/blms/33.1.41
  8. E. E. Enochs, A note on absolutely pure modules, Canad. Math. Bull. 19 (1976), 361-362 https://doi.org/10.4153/CMB-1976-054-5
  9. E. E. Enochs, Injective and fiat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189-209 https://doi.org/10.1007/BF02760849
  10. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000
  11. S. Jain, Flat and F P-injectivity, Proc. Amer. Math. Soc. 41 (1973), 437-442 https://doi.org/10.2307/2039110
  12. M. F. Jones, Flatness and f-projectivity of torsion-free modules and injective modules, Lecture Notes in Math. 951 (1982), 94-116 https://doi.org/10.1007/BFb0067327
  13. T. Y. Lam, Lectures on Modules and Rings; Springer-Verlag, New York-HeidelbergBerlin, 1999
  14. Z. K. Liu, Excellent extensions and homological dimensions, Comm. Algebra 22 (1994), no. 5, 1741-1745 https://doi.org/10.1080/00927879408824933
  15. B. Madox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155-158 https://doi.org/10.2307/2035245
  16. L. X. Mao and N. Q. Ding, FP-projective dimensions, Comm. Algebra 33 (2005), no. 4, 1153-1170 https://doi.org/10.1081/AGB-200053832
  17. K. R. Pinzon, Absolutely pure modules, University of Kentucky, Ph. D thesis, 2005
  18. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979
  19. B. Stenstrom, Coherent rings and FP-injective modules, J. London Math. Soc. 2 (1970), 323-329 https://doi.org/10.1112/jlms/s2-2.2.323
  20. J. Trlifaj, Covers, Envelopes, and Cotorsion Theories, Lecture notes for the workshop, 'Homological Methods in Module Theory'. Cortona, September 10-16, 2000
  21. M. Y. Wang, Some studies on ${\Pi}$-coherent rings, Proc. Amer. Math. Soc. 119 (1993), 71-76 https://doi.org/10.2307/2159825
  22. J. Xu, Flat Covers of Modules, Lecture Notes in Math. 1634, Springer-Verlag: BerlinHeidelberg-New York, 1996

Cited by

  1. PRECOVERS AND PREENVELOPES BY MODULES OF FINITE FGT-INJECTIVE AND FGT-FLAT DIMENSIONS vol.25, pp.4, 2010, https://doi.org/10.4134/CKMS.2010.25.4.497
  2. FGT-injective dimensions of Π-coherent rings and almost excellent extension vol.120, pp.2, 2010, https://doi.org/10.1007/s12044-010-0025-0