Gorenstein injective and strongly cotorsion modules

详细信息    查看全文

• 作者：Zhaoyong Huang (1)
  
• 刊名：Israel Journal of Mathematics
• 出版年：2013
• 出版时间：November 2013
• 年：2013
• 卷：198
• 期：1
• 页码：215-228
• 全文大小：265KB
• 参考文献：1. H. Bass, / On the ubiquity of Gorenstein rings, Mathematische Zeitschrift 82 (1963), 8鈥?8. CrossRef
  2. L. Bican, R. El Bashir and E. E. Enochs, / All modules have flat covers, The Bulletin of the London Mathematical Society 33 (2001), 385鈥?90. CrossRef
  3. J. E. Bj枚rk, / The Auslander condition on Noetherian rings, in / S茅minaire d鈥橝lg猫bre Paul Dubreil et Marie-Paul Malliavin, 39猫me Ann茅e, Paris, 1987/1988, Lecture Notes in Mathematics, Vol. 1404, Springer-Verlag, Berlin, 1989, pp.137鈥?73. CrossRef
  4. H. Cartan and S. Eilenberg, / Homological Algebra, Reprint of the 1956 original, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1999.
  5. N. Q. Ding and J. L. Chen, / The flat dimensions of injective modules, Manuscripta Mathematica 78 (1993), 165鈥?77. CrossRef
  6. E. E. Enochs, / Injective and flat covers, envelopes and resolvents, Israel Journal of Mathematics 39 (1981), 189鈥?09. CrossRef
  7. E. E. Enochs, / Flat covers and flat cotorsion modules, Proceedings of the American Mathematical Society 92 (1984), 179鈥?84. CrossRef
  8. E. E. Enochs and Z. Y. Huang, / Injective envelopes and (Gorenstein) flat covers, Algebras and Representation Theory 15 (2012), 1131鈥?145. CrossRef
  9. E. E. Enochs and O. M. G. Jenda, / Gorenstein injective and projective modules, Mathematische Zeitschrift 220 (1995), 611鈥?33. CrossRef
  10. E. E. Enochs and O. M. G. Jenda, / Relative Homological Algebra, De Gruyter Expositions in Mathematics, Vol. 30, Walter de Gruyter, Berlin, New York, 2000. CrossRef
  11. E. E. Enochs, O. M. G. Jenda and B. Torrecillas, / Gorenstein flat modules, Nanjing Daxue Xuebao. Shuxue Bannian Kan. Nanjing University. Journal. Mathematical Biquarterly 10 (1993), 1鈥?.
  12. D. J. Fieldhouse, / Character modules, Commentarii Mathematici Helvetici 46 (1971), 274鈥?76. CrossRef
  13. R. M. Fossum, P. A. Griffith and I. Reiten, / Trivial Extensions of Abelian Categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin, 1975.
  14. H. Holm, / Gorenstein homological dimensions, Journal of Pure and Applied Algebra 189 (2004), 167鈥?93. CrossRef
  15. Y. Iwanaga, / On rings with finite self-injective dimension II, Tsukuba Journal of Mathematics 4 (1980), 107鈥?13.
  16. K. Khashyarmanesh and Sh. Salarian, / On the rings whose injective hulls are flat, Proceedings of the American Mathematical Society 131 (2003), 2329鈥?335. CrossRef
  17. L. X. Mao and N. Q. Ding, / Envelopes and covers by modules of finite FP-injective and flat dimensions, Communications in Algebra 35 (2007), 833鈥?49. CrossRef
  18. J. Z. Xu, / Minimal injective and flat resolutions over Gorenstein rings, Journal of Algebra 175 (1995), 451鈥?77. CrossRef
  
• 作者单位：Zhaoyong Huang (1)
  
  1. Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu Province, P. R. China
  
• ISSN：1565-8511

文摘

By investigating the properties of some special covers and envelopes of modules, we prove that if R is a Gorenstein ring with the injective envelope of R R flat, then a left R-module is Gorenstein injective if and only if it is strongly cotorsion, and a right R-module is Gorenstein flat if and only if it is strongly torsionfree. As a consequence, we get that for an Auslander-Gorenstein ring R, a left R-module is Gorenstein injective (resp. flat) if and only if it is strongly cotorsion (resp. torsionfree).