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 $\mathrm{CE}(ℱ$). 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,...

Let $R$ be a commutative ring and $𝒞$ a semidualizing $R$-module. We investigate the relations between $𝒞$-flat modules and $𝒞$-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.