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.)