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I don't quite see why you aren't happy with the intuition that you give. It seems to me that it cleanly describes what the comultiplication is and how it arises.
I perhaps should have said, the more obvious generalisation of Krull dimension using prime ideals is not very good in general because "large" non-commutative rings can have very few prime ideals.
It is not automatic but I imagine will be true in any case that you might be interested in. Otherwise, you could replace R by a suitable quotient ring.
I think that does what I want. Thank you. It would be nice to have a reference that isn't the middle of a proof of something else entirely though. If anyone has any other suggestions they are very welcome.
