Now for the tricky part. The term *universal homeomorphism* is not yet defined, it only appears in definition 29.45.1 and in the context of schemes, despite its use in the title of section 10.46 (maybe that's a bit confusing, but this is not the main issue here).

It seems that throughout the Stacks Project, a great care is taken to not use the term *universal homeomorphism* for morphisms of rings. Instead, it is written in terms of morphisms on spectra (for instance in the proof of lemma 58.14.2), or with a sentence such as "*induces a universal homeomorphism on spectra*" (see for instance lemma 29.46.11).

So if we want to keep this consistency, I propose to first say that by lemma 10.45.4 (or by definition) is purely inseparable, so we can apply lemma 10.46.10 to find that has a unique prime ideal.

If this is not nitpicking for you, then I would suggest to also edit the very few remaining cases of universal homeomorphisms of rings in the Stacks Project: tags 29.47.6, 37.24.7, 109.39 (twice) and maybe 29.46.8.

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