Let A be an artin algebra. Then A has finite self-injective dimensions on both sides if and only if every finitely generated left ^-module has finite Gorenstein dimension. Two decades ago, Auslander and Bridger [2] showed that a commutative noetherian local ring A is a Gorenstein ring if and only if every finitely generated ^-module has finite Gorenstein dimension. In this paper, we will develop their arguments and apply obtained results to artin algebras. We will prove the following: Theorem.

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... et A be an artin algebra. Then inj dirn^ A = inj dim AA < oo ¿fand only if every finitely generated left A-module has finite Gorenstein dimension. In the following, we will work over a left and right noetherian ring A. We denote by ( )* both ,4-dual functors. Right ,4-modules are considered as left op-modules, where Aop stands for the opposite ring of A, and notations for left ^-modules are used also for left ^op-modules. Denote by mod ,4 the category of all finitely generated left ^-modules. For n > 0, a module M £ mod A is said to be an nth syzygy if there is an exact sequence 0 -> M -* Pn] -► • • ■ -► P0 in mod A with the P¡ projective. Let syz"^ be the class of all M £ mod .4 that are nth syzygies, and, for the sake of convenience, put syz A = mod/4 (i.e., every M £ mod.4 is considered as a Oth syzygy of itself). Also, a module M £ mod,4 is said to have reduced grade at least n, written reduced grade M ^ n, if Ext^(Af, A) = 0 for 1 < í < n. Note that reduced grade M > 1 for all M £ mod A. Finally, a module M e mod A is said to have Gorenstein dimension zero, written G-dim M = 0, if it is reflexive and reduced grade M -reduced grade M* = oo . Then, for n > 0, M is said to have Gorenstein dimension at most n , written G-dim M ^ n , if there is an exact sequence 0 -► Mn -► • • • -► M0 -> M -> 0 in mod A with G-dim Mi = 0 for 0 < / ^ n. Note that G-dim M < oo implies Ext^(Af, A) = 0 for i > G-dim M (see Auslander and Bridger [2] for details).