There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite rings, you get a compact ring; for example, the $p$-adic integers $\mathbb{Z}\_p$ are obtained as a limit of
$$
\cdots \twoheadrightarrow \mathbb{Z}/p^{n+1}\mathbb{Z} \twoheadrightarrow \mathbb{Z}/p^n\mathbb{Z}\twoheadrightarrow \cdots \twoheadrightarrow \mathbb{Z}/p\mathbb{Z}\twoheadrightarrow 0.
$$
Can every compact ring be obtained as a cofiltered limit of finite rings?


For a counterexample, a compact ring that is not totally disconnected would suffice. In the other direction, proving that such a ring has to be totally disconnected wouldn't suffice *a priori*: It would show the the additive *group* is profinite, but not that the ring is a cofiltered limit *of rings*.


Remark: By "compact," I consistently mean "compact Hausdorff."



Is every compact topological ring a profinite ring?
Every compact topological ring has "enough" open ideals and is thus profinite. See for example Sect. 5.1 in


Luis Ribes, Pavel Zalesski, *Profinite Groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge


