  In this paper we consider Artinian modules over power series rings endowed
with a Frobenius map. We describe a method for finding the set of all prime
annihilators of submodules which are preserved by the given Frobenius map and
on which the Frobenius map is not nilpotent. This extends the algorithm by Karl
Schwede and the first author, which solved this problem for submodules of the
injective hull of the residue field.
  The Matlis dual of this problem asks for the radical annihilators of
quotients of free modules by submodules preserved by a given Frobenius
near-splitting, and the same method solves this dual problem in the $F$-finite
case.
