Strongly Ƥ-projective Modules and Ƥ-projective Complexes
International Journal of Theoretical and Applied Mathematics
Volume 5, Issue 6, December 2019, Pages: 118-124
Received: Oct. 30, 2019; Accepted: Nov. 26, 2019; Published: Dec. 19, 2019
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Liang Yan, College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde, P. R. China
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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.
Strongly Ƥ-projective Module, Ƥ-projective Module, Ƥ-projective Complex
To cite this article
Liang Yan, Strongly Ƥ-projective Modules and Ƥ-projective Complexes, International Journal of Theoretical and Applied Mathematics. Vol. 5, No. 6, 2019, pp. 118-124. doi: 10.11648/j.ijtam.20190506.16
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F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, second ed., New York, Spring-verlag, 1992.
L. L. Avramov, H.-B. Foxby, Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71 (1991): 129-155.
J. L. Chen, P-Projective modules, Communications in Algebra, 24 (1996): 3, 821-831
E. E. Enochs, O. M. G. Jenda, Copure injective modules, Quaest. Math. 14 (1991) 401-409.
E. E. Enochs, O. M. G. Jenda, Copure injective resolutions, flat resolutions and dimensions, Comment. Math. 34 (1993) 203-211.
E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin, New York, 2000.
E. E. Enochs, L. Oyonarte, Covers, Envelopes and Cotorsion Theories. Nova Science Publishers, Inc. New York, 2002.
E. E. Enochs, S. Estrada, A. Iacob, Gorenstein projective and flat complexes over noetherian rings, Math. Nachr. (2012) 1-18.
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.
R. Gὅbel, J. Trlifaj, Approximations and Endomorphism Algebras of modules. Berlin-New York: Walter de Gruyter, 2006.
T. Y. Lam, Lectures on modules and rings, New York-Heidelberg-Berlin: Springer-Verlag, 1999.
L. Li, N. Q. Ding, G. Yang, covers and Envelopes by #-F Complexes. Communications in Algebra 39 (2011) 3253-3277.
L. X. Mao, N. Q. Ding, Relative copure injective and copure flat modules. Journal of Pure and Applied Algebra 208 (2007) 635-646.
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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