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It is well-known that compact Hausdorff topological unital rings are profinite. The proof generalises to (left or right) s-unital rings (i.e. rings such that for all $r\in R$ we have $r\in Rr$ or for all $r\in R$ we have $r\in rR$).

Is there a reference for this more general fact? Is there a further generalisation (i.e. an interesting class of rings, containing s-unital rings, for which compact Hausdorff implies profinite)?

(Note that this is not true for all rings, as given any compact Hausdorff abelian group $A$, we can endow $A$ with zero multiplication, making it a compact Hausdorff topological ring.)

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This is essentially answered in one of the answers to Is every compact topological ring a profinite ring?.

If a compact ring $R$ either admits no element $r\neq 0$ with $rR=0$ or the left-right dual condition then it is profinite. This is the condition that the multiplication map induces and embedding of $R$ into the endomorphisms of the Pontryagin dual of its additive group which is what you use to prove total disconnectedness.

See Thm 3 of On Compact Topologica Rings. by Hirotada Anzai https://projecteuclid.org/euclid.pja/1195573244

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  • $\begingroup$ Thanks, this is just what I was looking for. $\endgroup$ – tomasz Aug 16 at 14:31

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