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:
- $R_P$ is a discrete valuation ring for all height one primes $P$.
- 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.