DOI QR코드

DOI QR Code

ON REFLEXIVE MODULES OVER COMMUTATIVE RINGS

  • Geng, Yuxian ;
  • Ding, Nanqing
  • Received : 2013.09.03
  • Published : 2015.01.31

Abstract

Let R be a commutative ring and U an R-module. The aim of this paper is to study the duality between U-reflexive (pre)envelopes and U-reflexive (pre)covers of R-modules.

Keywords

reflexive module;(pre)envelope;(pre)cover

References

  1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.
  2. M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.
  3. R. G. Belshoff, Matlis reflexive modules, Comm. Algebra 19 (1991), no. 4, 1099-1118. https://doi.org/10.1080/00927879108824192
  4. R. G. Belshoff, Remarks on reflexive modules, covers, and envelopes, Beitrage Algebra Geom. 50 (2009), no. 2, 353-362.
  5. R. G. Belshoff, E. E. Enochs, and J. R. Garcia Rozas, Generalized Matlis duality, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1307-1312. https://doi.org/10.1090/S0002-9939-99-05130-8
  6. R. G. Belshoff and J. Z. Xu, Injective envelopes and flat covers of Matlis reflexive modules, J. Pure Appl. Algebra 79 (1992), no. 3, 205-215. https://doi.org/10.1016/0022-4049(92)90050-P
  7. W. Bruns and J. Herzog, Cohen-Macaulay Rings (revised edition), Advances in Math. 39, Cambridge Univ. Press, Cambridge, UK, 1996.
  8. L. W. Christensen, Gorenstein Dimension, Lecture Notes in Math. 1747, Springer-Verlag, Berlin, 2000.
  9. N. Q. Ding, On envelopes with the unique mapping property, Comm. Algebra 24 (1996), no. 4, 1459-1470. https://doi.org/10.1080/00927879608825646
  10. E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189-209. https://doi.org/10.1007/BF02760849
  11. E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Compact coGalois groups, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 2, 233-244. https://doi.org/10.1017/S0305004199004156
  12. E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Covering morphisms, Comm. Algebra 28 (2000), no. 8, 3823-3835. https://doi.org/10.1080/00927870008827060
  13. E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Are covering (enveloping) morphisms minimal?, Proc. Amer. Math. Soc. 128 (2000), no. 10, 2863-2868. https://doi.org/10.1090/S0002-9939-00-05339-9
  14. E. E. Enochs, J. R. Garcia Rozas, and L. Oyonarte, Finitely generated cotorsion modules, Proc. Edinb. Math. Soc. 44 (2001), no. 1, 143-152. https://doi.org/10.1017/S0013091599000590
  15. E. E. Enochs and Z. Y. Huang, Injective envelopes and (Gorenstein) flat covers, Algebr. Represent. Theory 15 (2012), no. 6, 1131-1145. https://doi.org/10.1007/s10468-011-9282-6
  16. 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
  17. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000.
  18. E. E. Enochs, O. M. G. Jenda, and J. Z. Xu, A generalization of Auslander's last theorem, Algebr. Represent. Theory 2 (1999), no. 3, 259-268. https://doi.org/10.1023/A:1009998109625
  19. M. Hashimoto and A. Shida, Some remarks on index and generalized Loewy length of a Gorenstein local ring, J. Algebra 187 (1997), no. 1, 150-162. https://doi.org/10.1006/jabr.1997.6770
  20. 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
  21. Z. Y. Huang, Proper resolutions and Gorenstein categories, J. Algebra 393 (2013), 142-169. https://doi.org/10.1016/j.jalgebra.2013.07.008
  22. Z. Y. Huang, Duality of preenvelopes and pure injective modules, Canad. Math. Bull. 57 (2014), no. 2, 318-325. https://doi.org/10.4153/CMB-2013-023-x
  23. D. A. Jorgensen and L. M. Sega, Independence of the total reflexivity conditions for modules, Algebr. Represent. Theory 9 (2006), no. 2, 217-226. https://doi.org/10.1007/s10468-005-0559-5
  24. W. V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349-1355. https://doi.org/10.1090/S0002-9939-1968-0237480-2
  25. W. M. Xue, Injective envelopes and flat covers of modules over a commutative ring, J. Pure Appl. Algebra 109 (1996), no. 2, 213-220. https://doi.org/10.1016/0022-4049(95)00085-2
  26. Y. Yoshino, Cohen-Macaulay approximations, In: Proc. symposium on representation theory of algebras (Izu, Japan, 1993) pp. 119-138 (in Japanese).

Acknowledgement

Supported by : NSFC, Jiangsu University of Technology of China