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

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.