Cotorsion dimensions relative to semidualizing modules

Let [Formula: see text] be a semidualizing [Formula: see text]-module, where [Formula: see text] is a commutative ring. We first introduce the definition of [Formula: see text]-cotorsion modules, and obtain the properties of [Formula: see text]-cotorsion modules. As applications, we give some new characterizations for perfect rings. Second, we study the Foxby equivalences between the subclasses of the Auslander class and that of the Bass class with respect to [Formula: see text]. Finally, we discuss [Formula: see text]-cotorsion dimensions and investigate the transfer properties of strongly [Formula: see text]-cotorsion dimensions under almost excellent extensions.

A noncommutative generalization of Auslander's last theorem

We show that every finitely generated leftR-module in the Auslander class over ann-perfect ringRhaving a dualizing module and admitting a Matlis dualizing module has a Gorenstein projective cover.