Property: countable

Definition: The underlying set of the ring is countable.

Reference(s):

(No citations retrieved.)

Metaproperties:

This property has the following metaproperties
  • passes to $eRe$ for any full idempotent $e$
  • passes to $eRe$ for any idempotent $e$
  • passes to subrings
  • passes to the center
  • stable under finite products
  • passes to matrix rings
  • Morita invariant
  • passes to localizations
  • passes to polynomial rings
  • passes to quotient rings
This property does not have the following metaproperties
  • stable under products (Counterexample: $R_{ 57 }$)
  • forms an equational class (counterexample needed)
Rings
Name
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$k[[x,y]]/(x^2,xy)$
$k[x;\sigma]/(x^2)$ (Artinian)
$T_\omega(\mathbb Q)$
2-dimensional uniserial domain
Akizuki's counterexample
Bergman's non-unit-regular subring
Bergman's primitive finite uniform dimension ring
Bergman's right-not-left primitive ring
Bergman's ring with IBN
Bergman's ring without IBN
Bergman's unit-regular ring
Camillo and Nielsen's McCoy ring
catenary, not universally catenary
Cohn's right-not-left free ideal ring
Cohn's Schreier domain that isn't GCD
Cozzens simple, left principal, right non-Noetherian domain
Cozzens' simple V-domain
DVR that is not N-2
Facchini's torch ring
Faith-Menal counterexample
Kaplansky's right-not-left hereditary ring
Kerr's Goldie ring with non-Goldie matrix ring
Kolchin's simple V-domain
Leavitt path algebra of an infinite bouquet of circles
Left-not-right pseudo-Frobenius ring
McGovern's commutative Zorn ring that isn't clean
Michler & Villamayor's right-not-left V ring
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
Nielsen's semicommutative ring that isn't McCoy
Noetherian ring that is not Grothendieck and not Nagata
non-$h$-local domain
Non-Artinian simple ring
Nonlocal endomorphism ring of a uniserial module
Page's left-not-right FPF ring
Pseudo-Frobenius, not quasi-Frobenius ring
Right-not-left simple injective ring
ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
Samuel's UFD having a non-UFD power series ring
Simple, connected, Noetherian ring with zero divisors
Simple, non-Artinian, von Neumann regular ring
Small's right hereditary, not-left semihereditary ring
Šter's counterexample showing "clean" is not Morita invariant
$2$-adic integers: $\mathbb Z_2$
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb C$: the field of complex numbers
$\mathbb H$: Hamilton's quaternions
$\mathbb Q[[x^2,x^3]]$
$\mathbb R$: 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
$\mathbb R[x]/(x^2)$
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty F_2$
$\widehat{\mathbb Z}$: 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
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$RCFM_\omega(\mathbb Q)$
10-adic numbers
Berberian's incompressible Baer ring
Bergman's exchange ring that isn't clean
Chase's left-not-right semihereditary ring
Clark's uniserial ring
field of $2$-adic numbers
Full linear ring of a countable dimensional right vector space
Goodearl's simple self-injective operator algebra
Goodearl's simple self-injective von Neumann regular ring
Interval monoid ring
Left-not-right uniserial domain
Local right-not-left Kasch ring
O'Meara's infinite matrix algebra
Osofsky's Type I ring
Puninski's triangular serial ring
Ring of holomorphic functions on $\mathbb C$
Square of a torch ring
Trivial extension torch ring
Šter's clean ring with non-clean corner rings
$\mathbb Q$: the field of rational numbers
$\mathbb Q(x)$: rational functions over the rational numbers
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z$: 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
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[x,x^{-1};\sigma]$
$k[x,y,z]/(xz,yz)$
$k[x,y]/(x^2, xy)$
$k[x,y]/(x^2-y^3)$
$k[x,y]_{(x,y)}/(x^2-y^3)$
$k[x;\sigma]/(x^2)$ (not right Artinian)
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$M_n(\mathbb Q)$
$M_n(F_2)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic closure of $F_2$
Algebraic integers
Basic ring of Nakayama's QF ring
Bass's right-not-left perfect ring
Bergman's example showing that "compressible" is not Morita invariant
Cohn's non-IBN domain
Countably infinite boolean ring
Custom Krull dimension valuation ring
Division algebra with no anti-automorphism
Division ring with an antihomomorphism but no involution
Eventually constant sequences in $\mathbb Z$
Field of algebraic numbers
Field of constructible numbers
Finitely cogenerated, not semilocal ring.
Grams' atomic domain which doesn't satisfy ACCP
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Henselization of $\Bbb Z_{(2)}$
Hochster's connected, nondomain, locally-domain ring
Hurwitz quaternions
Kasch not semilocal ring
Left-not-right Noetherian domain
Lipschitz quaternions
Malcev's nonembeddable domain
Nakayama's quasi-Frobenius ring that isn't Frobenius
Nielsen's right UGP, not left UGP ring
Noetherian domain that is not N-1
Non-symmetric $2$-primal ring
Osofsky's $32$ element ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
Progression free polynomial ring
Quasi-continuous ring that is not Ikeda-Nakayama
Ram's Ore extension ring
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Reversible non-symmetric ring
Right-not-left ACC on annihilators triangular ring
Right-not-left Artinian triangular ring
Right-not-left coherent ring
Right-not-left Kasch ring
Right-not-left Noetherian triangular ring
Right-not-left nonsingular ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Shepherdson's domain that is not stably finite
Varadarajan's left-not-right coHopfian ring
Legend
  • = has the property
  • = does not have the property
  • = information not in database