Nagata ring

In commutative algebra, an N-1 ring is an integral domain

whose integral closure in its quotient field is a finitely generated

A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, or a pseudo-geometric ring if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N-2 rings).

A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring,[1] but this concept is not used much.

Fields and rings of polynomials or power series in finitely many indeterminates over fields are examples of Japanese rings.

Another important example is a Noetherian integrally closed domain (e.g. a Dedekind domain) having a perfect field of fractions.

On the other hand, a principal ideal domain or even a discrete valuation ring is not necessarily Japanese.

The first example of a Noetherian domain that is not a Nagata ring was given by Akizuki (1935).

and an infinite degree field extension

Let the discrete valuation ring

be the ring of formal power series over

whose coefficients generate a finite extension of

is any formal power series not in

is not an N-1 ring (its integral closure is not a finitely generated module) so

is the subring of the polynomial ring

by adjoining inverses to all elements not in any of the ideals generated by some

is a 1-dimensional Noetherian domain that is not an N-1 ring, in other words its integral closure in its quotient field is not a finitely generated

has a cusp singularity at every closed point, so the set of singular points is not closed.