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More examples added at principal ideal domain.
A new subsection on torsionfree modules has been added to principal ideal domain. Totally standard material of course, but various spots around the nLab (e.g., proposition 2 of torsion subgroup) refer implicitly to it. At some point I plan to get around to the fundamental theorem on finitely generated modules.
Thanks, Todd, that's great.
Since you are pointing to what I had started to split off as Lazard's criterion at flat module, let's briefly think about how to reorganize the latter entry, currently it's a bit of a mess:
the characterization Explicitly in terms of identities that currently contains the idea of the proof of Lazard's criterion itself needs a bit of reasoning to derive from what I'd say is the the primary definition. I had started commenting on that in Equivalent characterizations.
Would you and Andrew and others who might care mind if I moved that whole section Explicitly in terms of identities to the Properties-subsection? We can still leave a prominent pointer to it at the very beginning of the Idea section, advizing the reader in need of an element-based characterization to skip to there. But currently this seems to me to be a bit out of place the way it sits in the Definition-section. In particular if we use it at the same time as the proof of Lazard's criterion.
Right, I put in that pointer because I was being a bit lazy. But since I was using the easy direction (a filtered colimit of frees is flat: easier to prove than a flat is a filtered colimit of frees), it probably makes sense to add just a few more lines, to make the proof self-contained. Because I generally dislike sending people off on mad goose-chases around the nLab, tracking down details of proofs.
I think I see what you mean about reorganization, but I’d like to think it over when I have more time (later today).
at principal ideal domain it used to say (since revision 4) that the entire holomorphic functions on the complex plane form an integral ideal domain.
It seems while every finitely generated ideal here is indeed principal, there are also non-finitely generated ideals and it is rather a Bezout domain.
I have edited it accordingly, but please check if I am being stupid here.
Huh, that’s weird. I do have a memory of writing some such thing, and the edit appears to be mine; I must have thought that I was transcribing something from Lang’s Algebra, but indeed all I see now from that book is an exercise that confirms what you say (every finitely generated ideal is principal, but it’s not even a UFD). Thanks for catching that.
Okay, thanks for confirming.
While we are at it: I ran across this coming from the introduction in Roosen’s book where it says that the whole function field analogy is already plausible from the just the bare fact that both $\mathbb{Z}$ and $\mathbb{F}_q[x]$ are principal ideal domains with finite group of units (and other finiteness properties). That already means that much of the theory will be the same, I think he says.
Now from the modern point of view complex curves are the “third column” of the function field analogy, and so I thought of the analogous statement there.
So how strong is being a Bezout domain really, from this point of view. Strong enough (maybe in addition with some other properties) to argue that just by this it is plausible that there ought to be a third column? I guess not, but it seems a natural question to wonder about here.
Added a very short remark about principal ideal domains in constructive mathematics.
Thanks. We can also try to fix the notion to make it work constructively; I put forward one naïve proposal.
added statement of Smith normal form (here)
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