  Let $R$ be a left and right Noetherian ring. We introduce the notion of the
torsionfree dimension of finitely generated $R$-modules. For any $n\geq 0$, 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 $n \geq 1$, we study the properties of the finitely
generated $R$-modules $M$ with $\Ext_R^i(M, R)=0$ for any $1\leq i \leq n$.
Then we investigate the relation between these properties and the
self-injective dimension of $R$.
