| Peer-Reviewed

Strongly Ƥ-projective Modules and Ƥ-projective Complexes

Received: 30 October 2019     Accepted: 26 November 2019     Published: 19 December 2019
Views:       Downloads:
Abstract

In this paper we first study the properties of Strongly Ƥ-projective modules, and obtain some equivalent conclusions about Strongly Ƥ-projective modules, it is proved that a finitely generated right R-module N is strongly Ƥ-projective if and only if Exti (N, R) = 0 for all i ≥ 1 over left noetherian and right perfect ring, a Ƥ-projective right R-module N is strongly Ƥ-projective if and only if the first syzygy of N is strongly Ƥ-projective. Then we extend the notion of Ƥ-projective modules to that of Ƥ-projective complexes. We study the relationships between Ƥ-projective complexes and Ƥ-projective modules, it is proved that a complex C is Ƥ-projective if and only if every Ci is Ƥ-projective for every integer i if and only if Ext1 (C, P) = 0 for every projective complex Ƥ if and only if for every exact sequence 0→AƤC→0 with Ƥ projective, A→Ƥ is a projective preenvelope of A. Some characterizations of Ƥ-projective complexes also obtained.

Published in International Journal of Theoretical and Applied Mathematics (Volume 5, Issue 6)
DOI 10.11648/j.ijtam.20190506.16
Page(s) 118-124
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Strongly Ƥ-projective Module, Ƥ-projective Module, Ƥ-projective Complex

References
[1] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, second ed., New York, Spring-verlag, 1992.
[2] L. L. Avramov, H.-B. Foxby, Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71 (1991): 129-155.
[3] J. L. Chen, P-Projective modules, Communications in Algebra, 24 (1996): 3, 821-831
[4] E. E. Enochs, O. M. G. Jenda, Copure injective modules, Quaest. Math. 14 (1991) 401-409.
[5] E. E. Enochs, O. M. G. Jenda, Copure injective resolutions, flat resolutions and dimensions, Comment. Math. 34 (1993) 203-211.
[6] E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin, New York, 2000.
[7] E. E. Enochs, L. Oyonarte, Covers, Envelopes and Cotorsion Theories. Nova Science Publishers, Inc. New York, 2002.
[8] E. E. Enochs, S. Estrada, A. Iacob, Gorenstein projective and flat complexes over noetherian rings, Math. Nachr. (2012) 1-18.
[9] J. R. Garcìa Rozas, Covers and Envelopes in the Category of Complexes of Modules. Boca Raton-London-New York-Washington, D. C.: CRC Press, 1999.
[10] R. Gὅbel, J. Trlifaj, Approximations and Endomorphism Algebras of modules. Berlin-New York: Walter de Gruyter, 2006.
[11] T. Y. Lam, Lectures on modules and rings, New York-Heidelberg-Berlin: Springer-Verlag, 1999.
[12] L. Li, N. Q. Ding, G. Yang, covers and Envelopes by #-F Complexes. Communications in Algebra 39 (2011) 3253-3277.
[13] L. X. Mao, N. Q. Ding, Relative copure injective and copure flat modules. Journal of Pure and Applied Algebra 208 (2007) 635-646.
[14] J. Trlifaj, Cover, Envelope, and Cotorsion Theories; Lecture notes for the workshop, Homological Methods in Module Theory, Cortona, September 10-16, 2000.
Cite This Article
  • APA Style

    Liang Yan. (2019). Strongly Ƥ-projective Modules and Ƥ-projective Complexes. International Journal of Theoretical and Applied Mathematics, 5(6), 118-124. https://doi.org/10.11648/j.ijtam.20190506.16

    Copy | Download

    ACS Style

    Liang Yan. Strongly Ƥ-projective Modules and Ƥ-projective Complexes. Int. J. Theor. Appl. Math. 2019, 5(6), 118-124. doi: 10.11648/j.ijtam.20190506.16

    Copy | Download

    AMA Style

    Liang Yan. Strongly Ƥ-projective Modules and Ƥ-projective Complexes. Int J Theor Appl Math. 2019;5(6):118-124. doi: 10.11648/j.ijtam.20190506.16

    Copy | Download

  • @article{10.11648/j.ijtam.20190506.16,
      author = {Liang Yan},
      title = {Strongly Ƥ-projective Modules and Ƥ-projective Complexes},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {5},
      number = {6},
      pages = {118-124},
      doi = {10.11648/j.ijtam.20190506.16},
      url = {https://doi.org/10.11648/j.ijtam.20190506.16},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20190506.16},
      abstract = {In this paper we first study the properties of Strongly Ƥ-projective modules, and obtain some equivalent conclusions about Strongly Ƥ-projective modules, it is proved that a finitely generated right R-module N is strongly Ƥ-projective if and only if Exti (N, R) = 0 for all i ≥ 1 over left noetherian and right perfect ring, a Ƥ-projective right R-module N is strongly Ƥ-projective if and only if the first syzygy of N is strongly Ƥ-projective. Then we extend the notion of Ƥ-projective modules to that of Ƥ-projective complexes. We study the relationships between Ƥ-projective complexes and Ƥ-projective modules, it is proved that a complex C is Ƥ-projective if and only if every Ci is Ƥ-projective for every integer i if and only if Ext1 (C, P) = 0 for every projective complex Ƥ if and only if for every exact sequence 0→A→Ƥ→C→0 with Ƥ projective, A→Ƥ is a projective preenvelope of A. Some characterizations of Ƥ-projective complexes also obtained.},
     year = {2019}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Strongly Ƥ-projective Modules and Ƥ-projective Complexes
    AU  - Liang Yan
    Y1  - 2019/12/19
    PY  - 2019
    N1  - https://doi.org/10.11648/j.ijtam.20190506.16
    DO  - 10.11648/j.ijtam.20190506.16
    T2  - International Journal of Theoretical and Applied Mathematics
    JF  - International Journal of Theoretical and Applied Mathematics
    JO  - International Journal of Theoretical and Applied Mathematics
    SP  - 118
    EP  - 124
    PB  - Science Publishing Group
    SN  - 2575-5080
    UR  - https://doi.org/10.11648/j.ijtam.20190506.16
    AB  - In this paper we first study the properties of Strongly Ƥ-projective modules, and obtain some equivalent conclusions about Strongly Ƥ-projective modules, it is proved that a finitely generated right R-module N is strongly Ƥ-projective if and only if Exti (N, R) = 0 for all i ≥ 1 over left noetherian and right perfect ring, a Ƥ-projective right R-module N is strongly Ƥ-projective if and only if the first syzygy of N is strongly Ƥ-projective. Then we extend the notion of Ƥ-projective modules to that of Ƥ-projective complexes. We study the relationships between Ƥ-projective complexes and Ƥ-projective modules, it is proved that a complex C is Ƥ-projective if and only if every Ci is Ƥ-projective for every integer i if and only if Ext1 (C, P) = 0 for every projective complex Ƥ if and only if for every exact sequence 0→A→Ƥ→C→0 with Ƥ projective, A→Ƥ is a projective preenvelope of A. Some characterizations of Ƥ-projective complexes also obtained.
    VL  - 5
    IS  - 6
    ER  - 

    Copy | Download

Author Information
  • College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde, P. R. China

  • Sections