  If $M$ is a nonzero finitely generated module over a commutative Noetherian
local ring $R$ such that $M$ has finite injective dimension and finite
Gorenstein dimension, then it follows from a result of Holm that $M$ has finite
projective dimension, and hence a result of Foxby implies that $R$ is
Gorenstein. We investigate whether the same conclusion holds for nonzero
finitely generated modules that have finite injective dimension and finite
reducing Gorenstein dimension, where the reducing Gorenstein dimension is a
finer invariant than the classical Gorenstein dimension, in general.
