Abstract
Let R be an associative ring with identity and F a class of R-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg F complexes and extend the basic properties of the class F to the class CE(F). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.
Similar content being viewed by others
References
F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13, Springer, New York, 1992.
H. Bass: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Am. Math. Soc. 95 (1960), 466–488.
D. Bennis, N. Mahdou: Global Gorenstein dimensions. Proc. Amer. Math. Soc. 138 (2010), 461–465.
H. Cartan, S. Eilenberg: Homological Algebra. Princeton Mathematical Series 19, Princeton University Press 15, Princeton, 1999.
S. U. Chase: Direct products of modules. Trans. Amer. Math. Soc. 97 (1960), 457–473.
T. J. Cheatham, D. R. Stone: Flat and projective character modules. Proc. Am. Math. Soc. 81 (1981), 175–177.
E. E. Enochs: Cartan-Eilenberg complexes and resolutions. J. Algebra 342 (2011), 16–39.
E. E. Enochs: Injective and flat covers, envelopes and resolvents. Isr. J. Math. 39 (1981), 189–209.
E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. Volume 1. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin, 2011.
E. E. Enochs, J. R. García Rozas: Tensor products of complexes. Math. J. Okayama Univ. 39 (1997), 17–39.
E. E. Enochs, J. A. López-Ramos: Kaplansky classes. Rend. Sem. Mat. Univ. Padova 107 (2002), 67–79.
D. J. Fieldhouse: Character modules. Comment. Math. Helv. 46 (1971), 274–276.
J. R. García Rozas: Covers and Envelopes in the Category of Complexes of Modules. Chapman & Hall/CRC Research Notes in Mathematics 407, Chapman and Hall/CRC, Boca Raton, 1999.
A. Iacob: DG-injective covers, #-injective covers. Commun. Algebra 39 (2011), 1673–1685.
J. J. Rotman: An Introduction to Homological Algebra. Universitext, Springer, Berlin, 2009.
J. -L. Verdier: Derived Categories of Abelian Categories. Astérisque 239. Société Mathématique de France, Paris, 1996. (In French.)
Z. Wang, Z. Liu: Complete cotorsion pairs in the category of complexes. Turk. J. Math. 37 (2013), 852–862.
J. Xu: Flat Covers of Modules. Lecture Notes in Mathematics 1634, Springer, Berlin, 1996.
G. Yang, L. Liang: Cartan-Eilenberg Gorenstein flat complexes. Math. Scand. 114 (2014), 5–25.
G. Yang, L. Liang: Cartan-Eilenberg Gorenstein projective complexes. J. Algebra Appl. 13 (2014), Article ID 1350068, 17 pages.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by National Natural Science Foundation of China (11261050, 11401475).
Rights and permissions
About this article
Cite this article
Zhai, X., Zhang, C. Cartan-Eilenberg projective, injective and flat complexes. Czech Math J 66, 151–167 (2016). https://doi.org/10.1007/s10587-016-0247-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-016-0247-0