  We obtain various characterizations of commutative Noetherian local rings
$(R, \fm)$ in terms of homological dimensions of certain finitely generated
modules. For example, we establish that $R$ is Gorenstein if the Gorenstein
injective dimension of the maximal ideal $\fm$ of $R$ is finite. Furthermore we
prove that $R$ must be regular if a single $\Ext_{R}^{n}(I,J)$ vanishes for
some integrally closed $\fm$-primary ideals $I$ and $J$ of $R$ and for some
integer $n\geq \dim(R)$. Along the way we observe that local rings that admit
maximal Cohen-Macaulay Tor-rigid modules are Cohen-Macaulay.
