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"

Reference(s):

(No citations retrieved.)

Metaproperties:

This property has the following metaproperties
  • passes to localizations
This property does not have the following metaproperties
  • stable under products (Counterexample: $R_{ 57 }$)
  • forms an equational class (counterexample needed)
  • passes to quotient rings (Counterexample: $R_{ 49 }$ is a homomorphic image of $R_{ 27 }$)
  • passes to subrings (Counterexample: $R_{ 131 }$)
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
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 $\mathbb Q$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x]/(x^2)$
$\mathbb Z/(n)$, $n$ divisible by two primes and a square
$\mathbb Z/(n)$, $n$ squarefree and not prime.
$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty F_2$
$\widehat{\mathbb Z}$: 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 $0$
$F_2[x,y]/(x,y)^2$
$k[[x,y]]/(x^2,xy)$
$k[x,y,z]/(xz,yz)$
$k[x,y]/(x^2, xy)$
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
10-adic numbers
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
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb R$: 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
$F_p(x)$
$k[x,y]/(x^2-y^3)$
$k[x,y]_{(x,y)}/(x^2-y^3)$
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 $\mathbb C^n$, $n>1$
Legend
  • = has the property
  • = does not have the property
  • = information not in database