Foxby equivalence over associative rings
We extend the definition of a semidualizing module to general associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule $C$. We then study the classes of $C$-flats, $C$-projectives, and $C$-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new even in the commutative, noetherian setting.