[Math] Integral domains equal to intersection of their height one localizations

ac.commutative-algebra

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)?

Best Answer

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.

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