Strongly <i>Ƥ</i>-projective Modules and <i>Ƥ</i>-projective Complexes

Liang Yan

doi:doi:10.11648/j.ijtam.20190506.16

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Strongly Ƥ-projective Modules and Ƥ-projective Complexes

Liang Yan

Published in International Journal of Theoretical and Applied Mathematics (Volume 5, Issue 6)

Received: 30 October 2019 Accepted: 26 November 2019 Published: 19 December 2019

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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.

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), 2024. Published by Science Publishing Group

Keywords

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

References

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[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.

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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

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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

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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

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@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}
}

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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  -

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Author Information

Liang Yan

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

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Plain Text BibTeX RIS

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

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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