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On GI-flat and GF-torsion modules

  • 1.

    School of Finance and Mathematics, West Anhui University, Luan 237012, China

  • 2.

    College of Mathematics, Qingdao University, Qingdao 266071, China

  • 3.

    School of Mathematical Sciences, Anhui University, Hefei 230039, China

Funds:  Supported by the Key Program of Excellent Youth Foundation of Higher Education in Anhui Province (2013SQRL071ZD), the National Natural Science Foundation of China (11126173).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.03.003
More Information
  • Author Bio:

    WANG Xiujian, male, born in 1982, PhD/lecturer. Research field: ring and algebra representation. E-mail:xjwang@wxc.edu.cn

  • Corresponding author: DU Xianneng
  • Received Date: 05 February 2013
  • Accepted Date: 03 March 2014
  • Rev Recd Date: 03 March 2014
  • Publish Date: 30 March 2015
    Abstract

  • Abstract

    Two classes of modules were studied: GI-flat and GF-torsion modules, where GI stands for Gorenstein injective modules and GF for Gorenstein flat modules. Two homological dimensions for a ring were investigated, the supremum of the flat dimension of Gorenstein injective modules and the supremum of the GF-torsion dimension of all modules. The relation between these classes of modules and the homological dimensions was also studied.

    Abstract

    Two classes of modules were studied: GI-flat and GF-torsion modules, where GI stands for Gorenstein injective modules and GF for Gorenstein flat modules. Two homological dimensions for a ring were investigated, the supremum of the flat dimension of Gorenstein injective modules and the supremum of the GF-torsion dimension of all modules. The relation between these classes of modules and the homological dimensions was also studied.
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    • References(17)
  • [1]
    Enochs E, Jenda O. On Gorenstein injective modules and projective modules[J]. Math Z, 1995, 220: 611-633.
    [2]
    Bennis D, Mahdou N. Strongly Gorenstein projective, injective and at modules[J]. J Pure Appl Algebra, 2007, 210: 437-445.
    [3]
    Gao Z H. Universal Gorenstein homological methods and its applications[D]. Chengdu: Sichuan Normal University, 2011.
    [4]
    Enochs E E, Jenda O M G. Copure injective resolutions, flat resolvents and dimensions[J]. Comment Math Univ Carolin, 1993, 34:203-211.
    [5]
    Gao Z H. On GI-flat modules and dimensions[J]. J Korean Math Soc, 2013, 50: 203-218.
    [6]
    Song W L, Huang Z Y. Gorenstein atness and injectivity over Gorenstein rings[J]. Science in China Series A: Mathematics, 2008, 51:215-218.
    [7]
    Chen J, Ding N. The flat dimensions of injective modules[J]. Manuscripta Math, 1993, 78:165-177.
    [8]
    Holm H. Gorenstein homological dimension[J]. J Pure Appl Algebra, 2004, 189: 167-193.
    [9]
    Tong W T. An Introduction to Homological Algebra[M]. Beijing: Higher Education Press, 1998.
    [10]
    Mahdou N, Tamekkante M, Yassemi S. On (strongly) Gorenstein Von Neumann regular rings[J]. Comm in Algebra, 2011, 39: 3 242-3 252.
    [11]
    Holm H. Rings with finite Gorenstein injective dimension[J]. Proc Amer Marh Soc, 2003, 132: 1 279-1 283.
    [12]
    Mahdou N, Tamekkante M. The orthogonal complement relative to the functor extension of the class of all Gorenstein at modules[J]. Adv Pure Appl Math, 2011, 2: 133-145.
    [13]
    Bennis D. Rings over which the class of Gorenstein at modules is closed under extensions[J]. Comm in Algebra, 2009, 37: 855-868.
    [14]
    Mahdou N, Tamekkante M. On (strongly) Gorenstein (semi)hereditary rings[J]. Arab J Sci Eng, 2011, 36: 431-440.
    [15]
    Bennis D, Mahdou N. Global Gorenstein dimensions[J] Proc Amer Math Soc, 2010, 138: 461-465.
    [16]
    Pan Q X, Zhu X L. GP-projective and GI-injective modules[J]. Math Notes, 2012, 91: 824-832.
    [17]
    Bennis D, Mahdou N. A generalization of strongly Gorenstein projective module[J]. J Algebra and Its Appl, 2009, 8: 219-227.
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    [1]
    Enochs E, Jenda O. On Gorenstein injective modules and projective modules[J]. Math Z, 1995, 220: 611-633.
    [2]
    Bennis D, Mahdou N. Strongly Gorenstein projective, injective and at modules[J]. J Pure Appl Algebra, 2007, 210: 437-445.
    [3]
    Gao Z H. Universal Gorenstein homological methods and its applications[D]. Chengdu: Sichuan Normal University, 2011.
    [4]
    Enochs E E, Jenda O M G. Copure injective resolutions, flat resolvents and dimensions[J]. Comment Math Univ Carolin, 1993, 34:203-211.
    [5]
    Gao Z H. On GI-flat modules and dimensions[J]. J Korean Math Soc, 2013, 50: 203-218.
    [6]
    Song W L, Huang Z Y. Gorenstein atness and injectivity over Gorenstein rings[J]. Science in China Series A: Mathematics, 2008, 51:215-218.
    [7]
    Chen J, Ding N. The flat dimensions of injective modules[J]. Manuscripta Math, 1993, 78:165-177.
    [8]
    Holm H. Gorenstein homological dimension[J]. J Pure Appl Algebra, 2004, 189: 167-193.
    [9]
    Tong W T. An Introduction to Homological Algebra[M]. Beijing: Higher Education Press, 1998.
    [10]
    Mahdou N, Tamekkante M, Yassemi S. On (strongly) Gorenstein Von Neumann regular rings[J]. Comm in Algebra, 2011, 39: 3 242-3 252.
    [11]
    Holm H. Rings with finite Gorenstein injective dimension[J]. Proc Amer Marh Soc, 2003, 132: 1 279-1 283.
    [12]
    Mahdou N, Tamekkante M. The orthogonal complement relative to the functor extension of the class of all Gorenstein at modules[J]. Adv Pure Appl Math, 2011, 2: 133-145.
    [13]
    Bennis D. Rings over which the class of Gorenstein at modules is closed under extensions[J]. Comm in Algebra, 2009, 37: 855-868.
    [14]
    Mahdou N, Tamekkante M. On (strongly) Gorenstein (semi)hereditary rings[J]. Arab J Sci Eng, 2011, 36: 431-440.
    [15]
    Bennis D, Mahdou N. Global Gorenstein dimensions[J] Proc Amer Math Soc, 2010, 138: 461-465.
    [16]
    Pan Q X, Zhu X L. GP-projective and GI-injective modules[J]. Math Notes, 2012, 91: 824-832.
    [17]
    Bennis D, Mahdou N. A generalization of strongly Gorenstein projective module[J]. J Algebra and Its Appl, 2009, 8: 219-227.
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