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We study prime ideals, prime modules, and associated primes of graded modulesover rings $S$ graded by a unique product monoid. We consider two situations indetail: (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 identitycomponent $R$ 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 thendescribe the set of associated prime ideals of an arbitrary induced module.  One of our main interests is to give necessary and sufficient conditions forprimeness, and to describe the associated primes, in the crossed product casewhen the action of the monoid is not an action by automorphisms; this includesthe case of a skew polynomial ring $R[x;\sigma]$ where $\sigma$ is anendomorphism of $R$.  At the end, we give some illustrative examples, several of which show thenecessity of the various hypotheses in our results.

Prime Modules and Associated Primes of Induced Modules over Rings Graded by Unique Product Monoids