Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$?

It is a standard fact that Krull domains, and thus noetherian normal rings, have this property. But Krull domains satisfy two additional properties, namely:

1. $R_P$ is a discrete valuation ring for all height one primes $P$.
2. Every non-zero element is contained in only a finite number of height one prime ideals.

What happens if I drop these two assumptions? Do I get anything new? Is this true for non-noetherian normal rings (I fear not)?

• A one-dimensional, local, integral domain has this property. So there are many examples, e.g., the local ring of $k[x,y]/\langle y^2 - x^3 \rangle$ at the maximal ideal $\langle x,y\rangle$. – Jason Starr Sep 1 '15 at 16:19

I cannot fully answer the question but some keywords:

• If you drop 1. what you get is called a weakly Krull domain.

• If you just weaken 1. to valuation ring what you get is called a generalized Krull domain.

There are a bunch of related notions. Thus, yes, you get something new and these types of rings got studied.

• I didn't know about "weakly Krull domains". Thank you, I'll take a look at this. – Uname Sep 1 '15 at 15:57
• After browsing through the literature I have to admit that I haven't seen any interesting non-obvious examples of weakly Krull domains. That's a bit annoying. – Uname Sep 3 '15 at 12:06
• I am not sure which examples you saw but one-dimensional noetherian domains are weakly Krull (but perhaps this is in the 'obvious' category). IIIRC Cohen-Macaulay domains are also weakly Krull. Some constructions can be found in projecteuclid.org/euclid.rmjm/1181069485 – user9072 Sep 3 '15 at 12:21
• Alright, quite a lot in there indeed, thanks. Do you happen to know whether there exist non-normal non-noetherian non-onedimensional rings with $R = \bigcap R_P$? Perhaps not all non- have to be satisfied at the same time. – Uname Sep 3 '15 at 12:24
• I would try to look at the subring of $C[x,y]$ generated by monomials $x^my^n$ with $m\leq n\sqrt 2$ or something like that. – Lev Borisov Feb 9 '16 at 22:35