Nagata ring

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