Maximal prime homomorphic images of mod-$p$ Iwasawa algebras

Let $k$ be a finite field of characteristic $p$, and $G$ a compact $p$-adic analytic group. Write $kG$ for the completed group ring of $G$ over $k$. In this paper, we describe the structure of the ring $kG\mathrm{/}P$, where $P$ is a minimal prime ideal of $kG$. We give an isomorphism between $kG\mathrm{/}P$ and a matrix ring with coefficients in the ring $(k^{\mathrm{\prime }}{G}^{\mathrm{\prime }}{)}_{\alpha }$, where $k^{\mathrm{\prime }}\mathrm{/}k$ is a finite field extension, $G^{\mathrm{\prime }}$ is a large subquotient of $G$ with no finite normal subgroups, and $(-{)}_{\alpha }$ is a “twisting” operation that preserves several desirable properties of the ring structure. We demonstrate an application of this isomorphism by setting up correspondences between certain ideals and subrings of $kG$ and those of $(k^{\mathrm{\prime }}{G}^{\mathrm{\prime }}{)}_{\alpha }$, and showing that these correspondences often preserve some useful properties, such as almost-faithfulness of an ideal, or control of an ideal by a closed normal subgroup.