  We answer a question of Celikbas, Dao, and Takahashi by establishing the
following characterization of Gorenstein rings: a commutative noetherian local
ring $(R,\mathfrak m)$ is Gorenstein if and only if it admits an integrally
closed $\mathfrak m$-primary ideal of finite Gorenstein dimension. This is
accomplished through a detailed study of certain test complexes. Along the way
we construct such a test complex that detect finiteness of Gorenstein
dimension, but not that of projective dimension.
