Relative Gorenstein global dimension

We study the relative Gorenstein projective global dimension of a ring with respect to a weakly Wakamatsu tilting module [Formula: see text]. We prove that this relative global dimension is finite if and only if the injective dimension of every module in Add[Formula: see text] and the [Formula: see text]-projective dimension of every injective module are both finite (indeed these three dimensions have a common upper bound). When RC satisfies some extra conditions we prove that the relative Gorenstein projective global dimension of [Formula: see text] is always bounded above by the [Formula: see text]-projective global dimension of [Formula: see text], these two dimensions being equal when the class of all [Formula: see text]-Gorenstein projective [Formula: see text]-modules is contained in the Bass class of [Formula: see text] relative to [Formula: see text]. Of course we also give the dual results concerning the relative Gorenstein injective global dimension.

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.

SEMIDUALIZING MODULES AND RELATED MODULES

In this paper, we prove that the Bass class [Formula: see text] with respect to a semidualizing bimodule C contains all FP-injective S-modules. We introduce the definition of C-FP-injective modules, and give some characterizations of right coherent rings in terms of the C-flat S-modules and C-FP-injective S op -modules. We discuss when every S-module has an C-flat preenvelope which is epic (or monic). In addition, we investigate the left and right [Formula: see text]-resolutions of R-modules by left derived functors Ext n(-, -) over a left Noetherian ring S. As applications, some new characterizations of left perfect rings are induced by these modules associated with C. A few classical results of these rings are obtained as corollaries.