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

Liang Yan

doi:doi:10.11648/j.ijtam.20190506.16

| Peer-Reviewed

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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), 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

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

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

Liang Yan

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

Download PDF

Sections

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

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