  Over a noetherian ring, it is a classic result of Matlis that injective
modules admit direct sum decompositions into injective hulls of quotients by
prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing
module, Gorenstein injective modules admit similar filtrations. We also
investigate Tor-modules of Gorenstein injective modules over such rings. This
extends work of Enochs and Huang over Gorenstein rings.
  Furthermore, we give examples showing the following: (1) the class of
Gorenstein injective $R$-modules need not be closed under tensor products, even
when $R$ is local and artinian; (2) the class of Gorenstein injective
$R$-modules need not be closed under torsion products, even when $R$ is a
local, complete hypersurface; and (3) the filtrations given in our main theorem
do not yield direct sum decompositions, even when $R$ is a local, complete
hypersurface.
