Journal of the
Korean Mathematical Society JKMS

Let $S$ and $R$ be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study $C$-weak flat and $C$-weak injective modules as a generalization of $C$-flat and $C$-injective modules (\cite{HW07}) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class $\mathcal{A}_C(R)$ and that of the Bass class $\mathcal{B}_C(S)$. Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in $\mathcal{A}_C(R)$ and $\mathcal{B}_C(S)$. Finally we consider an open question which is closely relative to the main results (\cite{EJL05}), and discuss the relationship between the Bass class $\mathcal{B}_C(S)$ and the class of Gorenstein injective modules.

Keywords: (faithfully) semidualizing bimodule, Auslander class, Bass class, $C$-weak injective module, $C$-weak flat module, Foxby equivalence, cover, preenvelope

MSC numbers: Primary 18G05, 16E30, 18G20