  We study prime ideals, prime modules, and associated primes of graded modules
over rings $S$ graded by a unique product monoid. We consider two situations in
detail: (a) the case where $S$ is strongly group-graded and (b) the case where
$S$ is a crossed product and the ideal or module is induced from the identity
component $R$ of $S$. We give explicit conditions for ideals and modules of $R$
to induce prime ideals of or prime modules over $S$ in these two cases. We then
describe the set of associated prime ideals of an arbitrary induced module.
  One of our main interests is to give necessary and sufficient conditions for
primeness, and to describe the associated primes, in the crossed product case
when the action of the monoid is not an action by automorphisms; this includes
the case of a skew polynomial ring $R[x;\sigma]$ where $\sigma$ is an
endomorphism of $R$.
  At the end, we give some illustrative examples, several of which show the
necessity of the various hypotheses in our results.
