DaRT - Ring Property 168

Rings

	Name
	$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb Z/(n)$, $n$ divisible by two primes and a square
	$\mathbb Z/(n)$, $n$ squarefree and not prime.
	$\mathbb Z[x]/(x^2-1)$
	$\mathbb Z[x_0, x_1,x_2,\ldots]$
	$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
	$\prod_{i=1}^\infty F_2$
	$\widehat{\mathbb Z}$: 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 $0$
	$k[[x,y]]/(x^2,xy)$
	10-adic numbers
	Akizuki's counterexample
	Clark's uniserial ring
	Eventually constant sequences in $\mathbb Z$
	Kerr's Goldie ring with non-Goldie matrix ring
	McGovern's commutative Zorn ring that isn't clean
	Nagata's normal ring that is not analytically normal
	non-$h$-local domain
	Pseudo-Frobenius, not quasi-Frobenius ring
	Ring of holomorphic functions on $\mathbb C$
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
	$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
	$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
	$\mathbb R[x_1, x_2,x_3,\ldots]$
	$\mathbb Z+x\mathbb Q[x]$
	$\mathbb Z[\sqrt{-5}]$
	$\mathbb Z[x]$
	$\mathbb Z[X]/(X^2,4X, 8)$
	$\mathbb Z[X]/(X^2,8)$
	$F_2[x,y]/(x,y)^2$
	$k[x,y,z]/(xz,yz)$
	$k[x,y]/(x^2, xy)$
	$k[x,y]/(x^2-y^3)$
	$k[x,y]_{(x,y)}/(x^2-y^3)$
	$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	Algebraic integers
	catenary, not universally catenary
	Cohn's Schreier domain that isn't GCD
	Countably infinite boolean ring
	Custom Krull dimension valuation 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
	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
	Perfect non-Artinian ring
	Perfect ring that isn't semiprimary
	Progression free polynomial 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 $\mathbb C^n$, $n>1$
	Samuel's UFD having a non-UFD power series 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
	$\mathbb Q[x]$
	$\mathbb R$: the field of real numbers
	$\mathbb R[[x]]$
	$\mathbb R[x]/(x^2)$
	$\mathbb Z$: the ring of integers
	$\mathbb Z/(2)$
	$\mathbb Z/(p)$, $p$ an odd prime
	$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
	$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
	$\mathbb Z[i]$: the Gaussian integers
	$\mathbb Z_S$, where $S=((2)\cup(3))^c$
	$\mathbb Z_{(2)}$
	$^\ast \mathbb R$: the field of hyperreal numbers
	$F_p(x)$
	Algebraic closure of $F_2$
	DVR that is not N-2
	Facchini's torch ring
	field of $2$-adic numbers
	Field of algebraic numbers
	Field of constructible numbers
	Henselization of $\Bbb Z_{(2)}$
	Noetherian ring that is not Grothendieck and not Nagata
	Osofsky's Type I ring
	Square of a torch ring
	Trivial extension torch ring

Legend

= has the property
= does not have the property
= information not in database

Property: FGC

Reference(s):

Metaproperties: