In basic homological algebra, the projective, injective and at dimensions of modules play an important and fundamental role. In this paper, the closely related Gorenstein projective, Gorenstein injective and Gorenstein at dimensions are studied.There is a variety of nice results about Gorenstein dimensions over special commutative noetherian rings; very often local Cohen-Macaulay rings with a dualizing module. These results are done by Avramov, Christensen, Enochs, Foxby, Jenda, Martsinkovsky and Xu among others. The aim of this paper is to generalize these results, and to give homological descriptions of the Gorenstein dimensions over arbitrary associative rings.Throughout this paper, R denotes a non-trivial associative ring. All modules are-if not speciÿed otherwise-left R-modules.When R is two-sided and noetherian, Auslander and Bridger [2] introduced in 1969 the G-dimension, G-dim R M , for every ÿnite, that is, ÿnitely generated, R-module M (see also [1] from 1966/67). They proved the inequality G-dim R M 6 pd R M , with equality G-dim R M = pd R M when pd R M is ÿnite. Furthermore they showed the generalized Auslander-Buchsbaum formula (sometimes known as the Auslander-Bridger formula) for the G-dimension.Over a general ring R, Enochs and Jenda deÿned in [9] a homological dimension, namely the Gorenstein projective dimension, Gpd R (−), for arbitrary (non-ÿnite) modules. It is deÿned via resolutions with (the so-called) Gorenstein projective modules.

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings.As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: ✩ Part of this work was done at MSRI during the spring semester of 2003, when the authors participated in the Program in Commutative Algebra. We thank the institution and program organizers for a very stimulating research environment.

We extend the definition of a semidualizing module to 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 in the commutative, noetherian setting.2000 Mathematics Subject Classification. 13D02, 13D07, 13D25, 16E05, 16E30.

A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom A (C, C) A in the derived category D(A).We show how each such module gives rise to three new homological dimensions which we call C-Gorenstein projective, C-Gorenstein injective, and C-Gorenstein flat dimension, and investigate the properties of these dimensions.

We introduce the notion of a duality pair and demonstrate how the left half of such a pair is "often" covering and preenveloping. As an application, we generalize a result by Enochs et al. on Auslander and Bass classes, and we prove that the class of Gorenstein injective modules-introduced by Enochs and Jenda-is covering when the ground ring has a dualizing complex.2000 Mathematics Subject Classification. 13D05, 13D07, 18G25.