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Cartan-Eilenberg projective, injective and flat complexes. (English) Zbl 1374.18024
Summary: Let $$R$$ be an associative ring with identity and $$\mathcal{F}$$ a class of $$R$$-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg $$\mathcal{F}$$ complexes and extend the basic properties of the class $$\mathcal{F}$$ to the class $$\text{CE}(\mathcal{F}$$). 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, $$\text{QF}$$ rings, semisimple rings, hereditary rings and perfect rings.
##### MSC:
 18G10 Resolutions; derived functors (category-theoretic aspects) 18G25 Relative homological algebra, projective classes (category-theoretic aspects) 18G35 Chain complexes (category-theoretic aspects), dg categories
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