For a left and right Noetherian ring $R$, we give some equivalent characterizations for $_RR$ satisfying the Auslander condition in terms of the flat (resp. injective) dimensions of the terms in a minimal injective coresolution (resp. flat resolution) of left $R$-modules. Furthermore, we prove that for an artin algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if the subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. As applications, we get some equivalent characterizations of Auslander-Gorenstein rings and Auslander-regular rings.