Property: N-2

Definition: A domain $R$ such that for every finite extension $L$ of its quotient field $K$, the integral closure of $R$ in $L$ is a finitely generated A-module. Also known as "Japanese rings"

	Name
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
	$\mathbb R[x_1, x_2,x_3,\ldots]$
	$\mathbb Z+x\mathbb Q[x]$
	$\mathbb Z[x_0, x_1,x_2,\ldots]$
	Akizuki's counterexample
	Base ring for $R_{187}$
	Cohn's Schreier domain that isn't GCD
	Custom Krull dimension valuation ring
	Grams' atomic domain which doesn't satisfy ACCP
	Henselization of $\Bbb Z_{(2)}$
	Kerr's Goldie ring with non-Goldie matrix ring
	Mori but not Krull domain
	Nagata's Noetherian infinite Krull dimension ring
	Nagata's normal ring that is not analytically normal
	Noetherian ring that is not Grothendieck and not Nagata
	non-$h$-local domain
	Osofsky's Type I ring
	Ring of holomorphic functions on $\mathbb C$
	Samuel's UFD having a non-UFD power series ring
	$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
	$\mathbb Q[x,y,z]/(xz,yz)$
	$\mathbb Q[x,y]/(x^2, xy)$
	$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
	$\mathbb R[x]/(x^2)$
	$\mathbb Z/(n)$, $n$ divisible by two primes and a square
	$\mathbb Z/(n)$, $n$ squarefree and not prime
	$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
	$\mathbb Z[X]/(X^2,4X, 8)$
	$\mathbb Z[X]/(X^2,8)$
	$\mathbb Z[x]/(x^2-1)$
	$\prod_{i=0}^\infty \mathbb Q$
	$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
	$\prod_{i=1}^\infty \mathbb Z/(2^i)$
	$\prod_{i=1}^\infty F_2$
	$\varinjlim \mathbb Q^{2^n}$
	$\widehat{\mathbb Z}$: the profinite completion of the integers
	$C([0,1])$, the ring of continuous real-valued functions on the unit interval
	$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
	$F_2[x,y]/(x,y)^2$
	$k[[x,y]]/(x^2,xy)$
	10-adic numbers
	2-truncated Witt vectors over $\Bbb F_2((t))$
	Clark's uniserial ring
	Countably infinite boolean ring
	DVR that is not N-2
	Eventually constant sequences in $\mathbb Z$
	Facchini's torch ring
	Finitely cogenerated, not semilocal ring
	Hochster's connected, nondomain, locally-domain ring
	Interval monoid ring
	Kasch not semilocal ring
	McGovern's commutative Zorn ring that isn't clean
	Noetherian domain that is not N-1
	Perfect non-Artinian ring
	Perfect ring that isn't semiprimary
	Progression free polynomial ring
	Pseudo-Frobenius, not quasi-Frobenius ring
	Quasi-continuous ring that is not Ikeda-Nakayama
	reduced $I_0$ ring that is not exchange
	reduced exchange ring which is not semiregular
	Square of a torch ring
	Trivial extension torch ring
	$2$-adic integers: $\mathbb Z_2$
	$\mathbb C$: the field of complex numbers
	$\mathbb Q$: the field of rational numbers
	$\mathbb Q(x)$: rational functions over the rational numbers
	$\mathbb Q[[X]]$
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb Q[x,y]/(x^2-y^3)$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
	$\mathbb Q[x]$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb R$: the field of real numbers
	$\mathbb R[[x]]$
	$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
	$\mathbb Z$: the ring of integers
	$\mathbb Z/(2)$
	$\mathbb Z/(p)$, $p$ an odd prime
	$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
	$\mathbb Z[\sqrt{-5}]$
	$\mathbb Z[i]$: the Gaussian integers
	$\mathbb Z[x]$
	$\mathbb Z_S$, where $S=((2)\cup(3))^c$
	$\mathbb Z_{(2)}$
	$^\ast \mathbb R$: the field of hyperreal numbers
	$F_p(x)$
	Algebraic closure of $F_2$
	Algebraic integers
	catenary, not universally catenary
	field of $2$-adic numbers
	Field of algebraic numbers
	Field of constructible numbers
	Nagata ring that not quasi-excellent
	ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$

Property: N-2

Reference(s):

Metaproperties: