In 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, (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 (Danilov 2001), 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 PID or even a DVR 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 (Akizuki 1935).
Famous quotes containing the word ring:
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Wearily, as one would turn to nod good-bye to Rochefoucauld,
If the street were time and he at the end of the street,
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—T.S. (Thomas Stearns)