In commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, an integral domain A
is called an N-1 ring if its integral closure in its quotient field is a finitely generated A module. It
is called a Japanese ring (or an N-2 ring) if for every
finite extension L of its quotient field K, the integral closure of A in L is a finitely generated A module (or equivalently a finite A-algebra). 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

Masayoshi Nagata

Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....

, (or a pseudo-geometric ring) if it is Noetherian

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

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

Fields and rings of polynomials

Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

 or power series in finitely many indeterminates over fields are examples of Japanese rings. Another important example is a Noetherian

Noetherian

In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...

 integrally closed domain

Integrally closed domain

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in the field of fractions of A is A itself...

 (e.g. a Dedekind domain

Dedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...

) having a perfect

Perfect field

In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...

 field of fractions

Field of fractions

In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...

. On the other hand, a PID

Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

 or even a DVR

Discrete valuation ring

In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...

 is not necessarily Japanese.

Any quasi-excellent ring is a Nagata ring, so in particular almost all Noetherian rings that occur in algebraic geometry are Nagata rings.
The first example of a Noetherian domain that is not a Nagata ring was given by Akizuki in .