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X-strongly Gorenstein modules

  • School of Mathematical Sciences, Anhui University, Hefei 230601, China

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https://doi.org/10.3969/j.issn.0253-2778.2020.02.007
  • Received Date: 29 May 2019
  • Accepted Date: 24 July 2019
  • Rev Recd Date: 24 July 2019
  • Publish Date: 28 February 2020
    Abstract

  • Abstract

    The notion of X-strongly Gorenstein projective module was defined. It was proved that a module is X-Gorenstein projective if and only if it is a direct summand of some X-strongly Gorenstein projective module. Furthermore, some basic properties of X-strongly Gorenstein modules were obtained. And some results about strongly Gorenstein modules were generalized or strengthened.

    Abstract

    The notion of X-strongly Gorenstein projective module was defined. It was proved that a module is X-Gorenstein projective if and only if it is a direct summand of some X-strongly Gorenstein projective module. Furthermore, some basic properties of X-strongly Gorenstein modules were obtained. And some results about strongly Gorenstein modules were generalized or strengthened.
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    • References(11)
  • [1]
    AUSLANDER M, BRIDGER M. Stable module theory[J]. Mem Amer Math Soc, 1969, 94:1-20.
    [2]
    ENOCHS E E, JENDA O M G. Gorenstein injective and projective modules[J]. Math Z, 1995, 220: 611-633.
    [3]
    CHRISTENSEN L W. Gorenstein Dimensions[M]. Berlin: Springer, 2000.
    [4]
    HOLM H. Gorenstein homological dimensions[J]. J Pure Appl Algebra, 2004, 189: 167-193.
    [5]
    BENNIS D, MAHDOU N. Gorensten global dimensions[J]. Pro Amer Math Soc, 2010, 138: 461-465.
    [6]
    LI Z W, ZHANG P. A construction of Gorenstein-projective modules[J]. J Algebra, 2010, 323:1802-1812.
    [7]
    BENNIS D, OUARGHI K. X-Gorenstein projective modules[J]. Inter Math Forum, 2010, 5(10): 487-491.
    [8]
    DING N Q, LI Y L, MAO L X. Strongly Gorenstein flat modules[J]. J Aust Math Soc, 2009, 86: 323-338.
    [9]
    BENNIS D, MAHDOU N. Strongly Gorenstein projective, injective, and flat modules[J]. J Pure App Algebra, 2007, 210: 435-445.
    [10]
    WANG J, XU X W, ZHAO Z B. X-Gorenstein projective dimensions[DB/OL]. [2019-05-01]. https://arxiv.org/abs/1801.09127.
    [11]
    佟文廷.同调代数引论[M]. 北京: 高等教育出版社，1998.)
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    [1]
    AUSLANDER M, BRIDGER M. Stable module theory[J]. Mem Amer Math Soc, 1969, 94:1-20.
    [2]
    ENOCHS E E, JENDA O M G. Gorenstein injective and projective modules[J]. Math Z, 1995, 220: 611-633.
    [3]
    CHRISTENSEN L W. Gorenstein Dimensions[M]. Berlin: Springer, 2000.
    [4]
    HOLM H. Gorenstein homological dimensions[J]. J Pure Appl Algebra, 2004, 189: 167-193.
    [5]
    BENNIS D, MAHDOU N. Gorensten global dimensions[J]. Pro Amer Math Soc, 2010, 138: 461-465.
    [6]
    LI Z W, ZHANG P. A construction of Gorenstein-projective modules[J]. J Algebra, 2010, 323:1802-1812.
    [7]
    BENNIS D, OUARGHI K. X-Gorenstein projective modules[J]. Inter Math Forum, 2010, 5(10): 487-491.
    [8]
    DING N Q, LI Y L, MAO L X. Strongly Gorenstein flat modules[J]. J Aust Math Soc, 2009, 86: 323-338.
    [9]
    BENNIS D, MAHDOU N. Strongly Gorenstein projective, injective, and flat modules[J]. J Pure App Algebra, 2007, 210: 435-445.
    [10]
    WANG J, XU X W, ZHAO Z B. X-Gorenstein projective dimensions[DB/OL]. [2019-05-01]. https://arxiv.org/abs/1801.09127.
    [11]
    佟文廷.同调代数引论[M]. 北京: 高等教育出版社，1998.)
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