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∏-COHERENT DIMENSIONS AND ∏-COHERENT RINGS

  • Mao, Lixin (DEPARTMENT OF BASIC COURSES NANJING INSTITUTE OF TECHNOLOGY)
  • Published : 2007.05.31

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

${\Pi}-coherent$ dimension;${\Pi}-coherent$ ring;FGT-injective module;FGT-flat module;FGT-injective dimension;preenvelope;precover

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. V. Camillo, Coherence for polynomial rings, J. Algebra 132 (1990), 72-76 https://doi.org/10.1016/0021-8693(90)90252-J
  3. 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
  4. N. Q. Ding, On envelopes with the unique mapping property, Comm. Algebra 24 (1996), no. 4, 1459-1470 https://doi.org/10.1080/00927879608825646
  5. N. Q. Ding and J. L. Chen, Relative coherence and preenvelopes, Manuscripta Math; 81 (1993), 243-262 https://doi.org/10.1007/BF02567857
  6. 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
  7. 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
  8. E. E. Enochs, Injective and fiat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189-209 https://doi.org/10.1007/BF02760849
  9. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000
  10. 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
  11. T. Y. Lam, Lectures on Modules and Rings; Springer-Verlag, New York-HeidelbergBerlin, 1999
  12. Z. K. Liu, Excellent extensions and homological dimensions, Comm. Algebra 22 (1994), no. 5, 1741-1745 https://doi.org/10.1080/00927879408824933
  13. B. Madox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155-158 https://doi.org/10.2307/2035245
  14. K. R. Pinzon, Absolutely pure modules, University of Kentucky, Ph. D thesis, 2005
  15. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979
  16. 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
  17. J. Trlifaj, Covers, Envelopes, and Cotorsion Theories, Lecture notes for the workshop, 'Homological Methods in Module Theory'. Cortona, September 10-16, 2000
  18. M. Y. Wang, Some studies on ${\Pi}$-coherent rings, Proc. Amer. Math. Soc. 119 (1993), 71-76 https://doi.org/10.2307/2159825
  19. J. Xu, Flat Covers of Modules, Lecture Notes in Math. 1634, Springer-Verlag: BerlinHeidelberg-New York, 1996
  20. L. Bonami, On the Structure of Skew Group Rings, Algebra Berichte 48, Verlag Reinhard Fisher, Munchen, 1984
  21. S. Jain, Flat and F P-injectivity, Proc. Amer. Math. Soc. 41 (1973), 437-442 https://doi.org/10.2307/2039110
  22. 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

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