# S-unital compact rings are profinite

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.)

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.