  We study Gorenstein dimension and grade of a module $M$ over a filtered ring
whose assosiated graded ring is a commutative Noetherian ring. An equality or
an inequality between these invariants of a filtered module and its associated
graded module is the most valuable property for an investigation of filtered
rings. We prove an inequality G-dim$M\leq{G-dim gr}M$ and an equality ${\rm
grade}M={\rm grade gr}M$, whenever Gorenstein dimension of ${\rm gr}M$ is
finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a
new viewpoint for studying filtered rings and modules. We apply these results
to a filtered ring with a Cohen-Macaulay or Gorenstein associated graded ring
and study a Cohen-Macaulay, perfect or holonomic module.
