TITLE: Residual finiteness for modules over group rings
QUESTION [9 upvotes]: Let $G$ be a finitely generated residually finite group and let $M$ be a finitely generated $\mathbb{Z}[G]$-module.
Question: Must $M$ be residually finite in the sense that for all nonzero $x \in M$, there exists some submodule $N$ of $M$ such that $x \notin N$ and $M/N$ is finite?
If this is not true in general, is it true if $G$ is also assumed to be nilpotent?

REPLY [4 votes]: Here is another example with different groups.  Formanek showed that the group ring over any field of a free product of non-trivial groups (and not both order 2) is primitive, has a faithful simple module.  That means that $\mathbb F_pG$ is primitive whenever $G=A\ast B$.  And of course $G$ is residually finite if both $A$ and $B$ are.  Obviously a faithful simple $\mathbb F_pG$-module is cyclic as a $\mathbb ZG$-module and since $G$ is infinite, it cannot be finite.  So this gives lots of examples including the free group on two generators.