Nagata ring

Remarks

Example: $\Z$ is a Nagata ring. It is Noetherian, and its quotients by primes are $\Z$ and the fields $\F_p$. A field $F$ is N-2, because the integral closure of $F$ in a finite extension $L$ is $L$, an $F$-module of finite dimension. The integral closure of $\Z$ in a number field $L$ is the ring of integers of $L$, a free $\Z$-module of rank $[L : \Q]$.

Non-example: there are one-dimensional Noetherian local domains whose integral closure in their own field of fractions is not a finitely generated module. Such a domain fails clause 2 at the prime $\mathfrak{p} = 0$.

A theorem of Nagata says that a Noetherian ring $R$ is a Nagata ring if and only if every finitely generated $R$-algebra that is a domain is N-2. Such rings are also called universally Japanese. Older texts call Nagata rings pseudo-geometric rings.