  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 $kG/P$, where $P$ is a
minimal prime ideal of $kG$. We give an isomorphism between $kG/P$ and a matrix
ring with coefficients in the ring $(k'G')_\alpha$, where $k'/k$ is a finite
field extension, $G'$ 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 $(k'G')_\alpha$, 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.
