"Being a UFD in general is not enough, but being a UFD in which 2, 3, and 5 are (co)prime is certainly sufficient for incommensurability of 3/2, 2, and 5/4 in t

"Being a UFD in general is not enough, but being a UFD in which 2, 3, and 5 are (co)prime is certainly sufficient for incommensurability of 3/2, 2, and 5/4 in the relevant sense." Ok, this seems like kind of a contrived condition though, compared to, say, the condition that 2, 3, and 5 are prime (which also suffices). Something that occurred to me after I made my last post (and possibly what you meant in the second paragraph?): perhaps the author didn't mean to imply that the UFD property is useful for characterizing when you have incommensurability in different rings, but rather that it's relevant because it's used as a step in the proof that these ratios are incommensurable in the case of Z. This makes sense to me but in my opinion it wouldn't hurt if the article were more explicit on this point. I'm not familiar with a commonly-used definition of UFD for which the ring of integers mod 7 is not a UFD. Could you please point me to a reference that contains this alternate definition of UFD you are referring to? The definition I am using is the one in Abstract Algebra by Dummit & Foote, which happens to agree with the definition on wikipedia at the time of this posting.