Nagata ring
In commutative algebra, an N-1 ring is an integral domain formula_1 whose integral closure in its quotient field is a finitely generated formula_1-module. It is called a Japanese ring (or an N-2 ring) if for every finite extension formula_3 of its quotient field formula_4, the integral closure of formula_1 in formula_3 is a finitely generated formula_1-module (or equivalently a finite formula_1-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, 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 , but this concept is not used much. Examples. 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. 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 . Here is an example of a discrete valuation ring that is not a Japanese ring. Choose a prime formula_9 and an infinite degree field extension formula_4 of a characteristic formula_9 field formula_12, such that formula_13. Let the discrete valuation ring formula_14 be the ring of formal  power series over formula_4 whose coefficients generate a finite extension of formula_12. If formula_17 is any formal power series not in formula_14 then the ring formula_19 is not an N-1 ring (its integral closure is not a finitely generated module) so formula_14 is not a Japanese ring. If formula_14 is the subring of the polynomial ring formula_22 in infinitely many generators generated by the squares and cubes of all generators, and formula_23 is obtained from formula_14 by adjoining inverses to all elements not in any of the ideals generated by some formula_25, then formula_23 is a Noetherian domain that is  not an N-1 ring, in other words its integral closure in its quotient field is not a finitely generated formula_23-module. Also formula_23 has a cusp singularity at every closed point, so the set of singular points is not closed.