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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."

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    $\begingroup$ ... and by "ring" you mean something associative with a unit, since the zero multiplication law would make any non-profinite compact group a counterexample. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 9, 2010 at 6:49
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    $\begingroup$ Assume that the compact ring $A$ is an integral domain. Then its field of fractions $K$ is a locally compact field, which have all been classified. $\endgroup$
    – Chandan Singh Dalawat
    Commented Dec 9, 2010 at 6:57
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    $\begingroup$ Google claims this is true, and is proved for example in theorem 26.10 of the book "Locally compact groups" by Markus Stroppel. $\endgroup$
    – Gjergji Zaimi
    Commented Dec 9, 2010 at 7:21
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    $\begingroup$ Andreas: I wrote so, to indicate that the proof appears somewhere but I can not see it at the moment. Perhaps someone who knows the proof can write an answer with a quick sketch/main idea etc. Plus it appears that the result is old and possibly due to Kaplansky, so I'd like to research that a little more. $\endgroup$
    – Gjergji Zaimi
    Commented Dec 9, 2010 at 9:32
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    $\begingroup$ @ChandanSinghDalawat The paper "Compact rings" (Mathematische Annalen 145 (Feb 1962) 52-63) by Seth Warner is concluded "with an example of a compact subdomain of a topological field which is not a subdomain of any locally compact field". $\endgroup$
    – Alex Ravsky
    Commented Sep 26, 2023 at 10:32

2 Answers 2

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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

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The earliest reference I could find to the fact that compact Hausdorff rings are profinite (objects in the category of topological rings) is in Johnstone's "Stone Spaces" (VI.4.11 on page 266); in the bibliographical notes on page 269 it is attributed to Kaplansky, "Topological Rings" (Amer. J. Math. 69 (1947) 153-183). Interestingly, Kaplansky himself refers to Otobe ("On quasi-evaluations of compact rings," Proceedings of the Imperial Academy of Tokyo, 20 (1944), pp. 278-282) who in turn refers to Anzai ("On compact topological rings", same Proceedings, 19 (1943), 613-616); and Anzai says he owes the proof of the profiniteness part to Nakayama.

Decided to mention all this since none of the references in answers or comments seem to contain these attributions, except for the comment by Gjergji Zaimi "...it appears that the result is old and possibly due to Kaplansky, so I'd like to research that a little more."

PS Concerning units - actually Anzai does not require it, the only thing needed is that the zero is the sole element annihilating the whole ring.

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