  We prove that for a left and right Noetherian ring $R$, $_RR$ satisfies the
Auslander condition if and only if so does every flat left $R$-module, if and
only if the injective dimension of the $i$th term in a minimal flat resolution
of any injective left $R$-module is at most $i-1$ for any $i \geq 1$, if and
only if the flat (resp. injective) dimension of the $i$th term in a minimal
injective coresolution (resp. flat resolution) of any left $R$-module $M$ is at
most the flat (resp. injective) dimension of $M$ plus $i-1$ for any $i \geq 1$,
if and only if the flat (resp. injective) dimension of the injective envelope
(resp. flat cover) of any left $R$-module $M$ is at most the flat (resp.
injective) dimension of $M$, and if and only if any of the opposite versions of
the above conditions hold true. Furthermore, we prove that for an Artinian
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.
