Let
be a commutative Noetherian local
ring with identity. In this paper we investigate the existence of
Cohen-Macaulay modules with special annihilators applying the finite
filtration of submodules of finitely generated modules. As an
application, we deduce several consequences concerning the
associated primes of the quotient rings of the
-adic
completion of
. Also, we shall prove some results concerning the
vanishing of extension functors.