Let $R$ be a left Noetherian ring (if you prefer you can just think to $R$ as a skew field, I'll be happy with an answer under that hypothesis) and $G$ be a polycyclic-by-finite (or, if you prefer Abelian-by-finite) group. Take a crossed product $R*G$, a left ideal $I$ and a subgroup $H$ of $G$. $H$ is said to control $I$ if $R*G(I\cap R*H)=I$.

My questions are the following:

(1) is it true that the minimal prime ideals of R*G are controlled by the finite subgroups of $G$?

(2) if the answer to (1) is positive, is it true that for a minimal prime ideal $\mathfrak p\leq R*G$, there exists a finite subgroup $H$ of $G$ and a minimal prime ideal $\mathfrak q\leq R*H$ such that $(R*G)\mathfrak q=\mathfrak p$?

A positive answer to (1) in the case of $R[G]$ with $G$ polycyclic is given in Corollary 22 of [J.E.Roseblade, Prime ideals in group rings of polycyclic groups, 1977]. There seems to be a wide literature on prime ideals of crossed products of group rings, I would really appreciate some precise references. Furthermore, in case the answers to my questions are positive, it would be very helpful for me to have some direct argument, i.e., not following from a much deeper result requiring heavy machinery.


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