  $\tau$-rigid modules are essential in the $\tau$-tilting theory introduced by
Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for
Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms
of $\tau$-rigid modules. We show that every indecomposable module over iterated
tilted algebras of Dynkin type is $\tau$-rigid. Finally, we give a
$\tau$-tilting theorem on homological dimension which is an analog to that of
classical tilting modules.
