Let R be a left and right Noetherian ring. We introduce the notion of the
torsionfree dimension of finitely generated R-modules. For any n≥0, we
prove that R is a Gorenstein ring with self-injective dimension at most n
if and only if every finitely generated left R-module and every finitely
generated right R-module have torsionfree dimension at most n, if and only
if every finitely generated left (or right) R-module has Gorenstein dimension
at most n. For any n≥1, we study the properties of the finitely
generated R-modules M with \Ext_R^i(M, R)=0 for any 1≤i≤n.
Then we investigate the relation between these properties and the
self-injective dimension of R.Comment: 16 pages. The proof of Lemma 3.8 is modified. It has been accepted
for publication in Osaka Journal of Mathematic