# 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)
Rings
Name
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
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
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]$ 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 $0$
$F_2[x,y]/(x,y)^2$
$k[[x,y]]/(x^2,xy)$
2-truncated Witt vectors over $\Bbb F_2((t))$
Akizuki's counterexample
Algebraic integers
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 $\mathbb Z$
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
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 $\mathbb C$
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
$\mathbb Q[x,x^{-1}]$: 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
$^\ast \mathbb R$: 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