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Article

MSC: 18G10, 18G25, 18G35
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Keywords:
Cartan-Eilenberg projective complex; Cartan-Eilenberg injective complex; Cartan-Eilenberg flat complex
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 ${\rm CE}(\mathcal {F}$). 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, $\rm QF$ rings, semisimple rings, hereditary rings and perfect rings.
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