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Here is another example with easier 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.

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.

Here is another example with easier groups. Formanek showed that the group ring over any field of a free product of non-trivial groups 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.

Here is another example with easier 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.

Here is another example with easier groups. Formanek showed that the group ring over any field of a free product of non-trivial groups 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.

Here is another example with easier groups. Formanek showed that the group ring over any field of a free product of non-trivial groups 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.