Property: valuation domain

Definition: A commutative integral domain whose ideals are linearly ordered. (In some places also called a "valuation ring")

Reference(s):

(No citations retrieved.)

Metaproperties:

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)

	Name
	$2$-adic integers: $\mathbb Z_2$
	$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
	$\mathbb C$: the field of complex numbers
	$\mathbb Q$: the field of rational numbers
	$\mathbb Q(x)$: rational functions over the rational numbers
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb Q[x,y,z]/(xz,yz)$
	$\mathbb Q[x,y]/(x^2, xy)$
	$\mathbb Q[x,y]/(x^2-y^3)$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
	$\mathbb Q[x]$
	$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb R$: the field of real numbers
	$\mathbb R[[x]]$
	$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
	$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
	$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
	$\mathbb R[x]/(x^2)$
	$\mathbb R[x_1, x_2,x_3,\ldots]$
	$\mathbb Z$: the ring of integers
	$\mathbb Z+x\mathbb Q[x]$
	$\mathbb Z/(2)$
	$\mathbb Z/(n)$, $n$ divisible by two primes and a square
	$\mathbb Z/(n)$, $n$ squarefree and not prime
	$\mathbb Z/(p)$, $p$ an odd prime
	$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
	$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
	$\mathbb Z[\sqrt{-5}]$
	$\mathbb Z[i]$: the Gaussian integers
	$\mathbb Z[x]$
	$\mathbb Z[X]/(X^2,4X, 8)$
	$\mathbb Z[X]/(X^2,8)$
	$\mathbb Z[x]/(x^2-1)$
	$\mathbb Z[x_0, x_1,x_2,\ldots]$
	$\mathbb Z_S$, where $S=((2)\cup(3))^c$
	$\mathbb Z_{(2)}$
	$\prod_{i=0}^\infty \mathbb Q$
	$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
	$\prod_{i=1}^\infty \mathbb Z/(2^i)$
	$\prod_{i=1}^\infty F_2$
	$\varinjlim \mathbb Q^{2^n}$
	$\widehat{\mathbb Z}$: the profinite completion of the integers
	$^\ast \mathbb R$: the field of hyperreal numbers
	$C([0,1])$, 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 $0$
	$F_2[x,y]/(x,y)^2$
	$F_p(x)$
	$k[[x,y]]/(x^2,xy)$
	10-adic numbers
	2-truncated Witt vectors over $\Bbb F_2((t))$
	Akizuki's counterexample
	Algebraic closure of $F_2$
	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
	DVR that is not N-2
	Eventually constant sequences in $\mathbb Z$
	Facchini's torch ring
	field of $2$-adic numbers
	Field of algebraic numbers
	Field of constructible numbers
	Finitely cogenerated, not semilocal ring
	Grams' atomic domain which doesn't satisfy ACCP
	Henselization of $\Bbb Z_{(2)}$
	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
	Nagata's normal ring that is not analytically normal
	Noetherian domain that is not N-1
	Noetherian ring that is not Grothendieck and not Nagata
	non-$h$-local domain
	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

Name

$2$-adic integers: $\mathbb Z_2$

$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$

$\mathbb C$: the field of complex numbers

$\mathbb Q$: the field of rational numbers

$\mathbb Q(x)$: rational functions over the rational numbers

$\mathbb Q[[x^2,x^3]]$

$\mathbb Q[\mathbb Q]$

$\mathbb Q[x,x^{-1}]$: Laurent polynomials

$\mathbb Q[x,y,z]/(xz,yz)$

$\mathbb Q[x,y]/(x^2, xy)$

$\mathbb Q[x,y]/(x^2-y^3)$