Property: Euclidean domain

Definition: A domain which has a Euclidean valuation

Reference(s):

  • N. Jacobson. Basic algebra I. (2012) @ Section 2.16

Metaproperties:

This property has the following metaproperties
  • passes to localizations
This property does not have the following metaproperties
  • passes to quotient rings (Counterexample: $R_{ 49 }$ is a homomorphic image of $R_{ 27 }$)
  • passes to subrings (Counterexample: $R_{ 6 }$ is a subring of $R_{ 101 }$)
  • stable under finite products (Counterexample: $R_{ 9 }$)
  • stable under products (counterexample needed)
  • forms an equational class (counterexample needed)
  • passes to polynomial rings (Counterexample: $R_{ 207 }$ is the polynomial ring of $R_{ 7 }$)
Rings
Name
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
Base ring for $R_{187}$
DVR that is not N-2
Henselization of $\Bbb Z_{(2)}$
Nagata's normal ring that is not analytically normal
Noetherian ring that is not Grothendieck and not Nagata
non-$h$-local domain
Principal ideal domain that is not Nagata
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2-1)$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
$\mathbb R[x]/(x^2)$
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\mathbb Z+x\mathbb Q[x]$
$\mathbb Z/(n)$, $n$ divisible by two primes and a square
$\mathbb Z/(n)$, $n$ squarefree and not prime
$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
$\mathbb Z[\sqrt{-5}]$
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$\varinjlim \mathbb Q^{2^n}$
$\widehat{\mathbb Z}$: 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 $\Bbb F_2((t))$
ACCP ring with non-ACCP polynomial ring
ACCP ring with non-ACCP power-series
Akizuki's counterexample
Algebraic integers
Atomic domain with nonatomic polynomial ring
catenary, not universally catenary
Clark's uniserial ring
Cohn's Schreier domain that isn't GCD
Countably infinite boolean ring
Custom Krull dimension valuation ring
Eventually constant sequences in $\mathbb Q$
Eventually constant sequences in $\mathbb Z$
Facchini's torch ring
Finitely cogenerated, not semilocal ring
Grams' atomic domain which doesn't satisfy ACCP
Hochster's connected, nondomain, locally-domain ring
Interval monoid ring
Kasch not semilocal ring
Kerr's Goldie ring with non-Goldie matrix ring
McGovern's commutative Zorn ring that isn't clean
Mori but not Krull domain
Nagata ring that not quasi-excellent
Nagata's Noetherian infinite Krull dimension ring
Noetherian domain that is not N-1
Non-ACCP polynomial ring
Non-ACCP power-series
Non-atomic polynomial ring
Noncoherent product of coherent rings
Osofsky's Type I ring
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
ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
Ring of holomorphic functions on $\mathbb C$
Ring of integer valued polynomials over the rationals
Samuel's UFD having a non-UFD power series ring
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
$\mathbb Q[[X]]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x]$
$\mathbb R$: the field of real numbers
$\mathbb R[[x]]$
$\mathbb Z$: the ring of integers
$\mathbb Z/(2)$
$\mathbb Z/(p)$, $p$ an odd prime
$\mathbb Z[i]$: the Gaussian integers
$\mathbb Z_{(2)}$
$^\ast \mathbb R$: the field of hyperreal numbers
$F_p(x)$
Algebraic closure of $F_2$
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Legend
  • = has the property
  • = does not have the property
  • = information not in database