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Type: cardinality

Definition: The cardinality of the underlying set of the ring.

Displaying only commutative rings. Show all rings

Name Dimension
$\mathbb Q$: the field of rational numbers $\aleph_0$
$\mathbb Q(x)$: rational functions over the rational numbers $\aleph_0$
$\mathbb Q[\mathbb Q]$ $\aleph_0$
$\mathbb Q[X,Y]_{(X,Y)}$ $\aleph_0$
$\mathbb Q[x]$ $\aleph_0$
$\mathbb Q[x_1, x_2,\ldots, x_n]$ $\aleph_0$
$\mathbb Z$: the ring of integers $\aleph_0$
$\mathbb Z+x\mathbb Q[x]$ $\aleph_0$
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$ $\aleph_0$
$\mathbb Z[\sqrt{-5}]$ $\aleph_0$
$\mathbb Z[i]$: the Gaussian integers $\aleph_0$
$\mathbb Z[x]$ $\aleph_0$
$\mathbb Z[x]/(x^2-1)$ $\aleph_0$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$ $\aleph_0$
$\mathbb Z_{(2)}$ $\aleph_0$
$F_p(x)$ $\aleph_0$
$k[x,y,z]/(xz,yz)$ $\aleph_0$
$k[x,y]/(x^2, xy)$ $\aleph_0$
$k[x,y]/(x^2-y^3)$ $\aleph_0$
$k[x,y]_{(x,y)}/(x^2-y^3)$ $\aleph_0$
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$ $\aleph_0$
Algebraic integers $\aleph_0$
Countably infinite boolean ring $\aleph_0$
Custom Krull dimension valuation ring $\aleph_0$
Field of algebraic numbers $\aleph_0$
Field of constructible numbers $\aleph_0$
Finitely cogenerated, not semilocal ring. $\aleph_0$
Grams' atomic domain which doesn't satisfy ACCP $\aleph_0$
Henselization of $\Bbb Z_{(2)}$ $\aleph_0$
Hochster's connected, nondomain, locally-domain ring $\aleph_0$
Kasch not semilocal ring $\aleph_0$
Nagata's Noetherian infinite Krull dimension ring $\aleph_0$
Perfect non-Artinian ring $\aleph_0$
Perfect ring that isn't semiprimary $\aleph_0$
Quasi-continuous ring that is not Ikeda-Nakayama $\aleph_0$
reduced $I_0$ ring that is not exchange $\aleph_0$
reduced exchange ring which is not semiregular $\aleph_0$
$2$-adic integers: $\mathbb Z_2$ $\mathfrak c$
$\mathbb Q[[x^2,x^3]]$ $\mathfrak c$
$\mathbb R[[x]]$ $\mathfrak c$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$ $\mathfrak c$
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$ $\mathfrak c$
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions $\mathfrak c$
$\mathbb R[x]/(x^2)$ $\mathfrak c$
$\mathbb R[x_1, x_2,x_3,\ldots]$ $\mathfrak c$
$\prod_{i=1}^\infty F_2$ $\mathfrak c$
Clark's uniserial ring $\mathfrak c$
field of $2$-adic numbers $\mathfrak c$
Interval monoid ring $\mathfrak c$
Ring of holomorphic functions on $\mathbb C$ $\mathfrak c$
$\mathbb C$: the field of complex numbers $\mathfrak{c}$
$\mathbb R$: the field of real numbers $\mathfrak{c}$
$^\ast \mathbb R$: the field of hyperreal numbers $\mathfrak{c}$
$\mathbb Z/(n)$, $n$ divisible by two primes and a square $n$
$\mathbb Z/(n)$, $n$ squarefree and not prime. $n$
$\mathbb Z/(p)$, $p$ an odd prime $p$
$\mathbb Z/(p^k)$, $p$ a prime, $k>1$ $p^k$
$\mathbb Z/(2)$ 2
$\mathbb Z[X]/(X^2,4X, 8)$ 32
$\mathbb Z[X]/(X^2,8)$ 64
$F_2[x,y]/(x,y)^2$ 8
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\widehat{\mathbb Z}$: 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
Algebraic closure of $F_2$
catenary, not universally catenary
Cohn's Schreier domain that isn't GCD
DVR that is not N-2
Eventually constant sequences in $\mathbb Z$
Facchini's torch 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 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
Progression free polynomial ring
Pseudo-Frobenius, not quasi-Frobenius ring
ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
Samuel's UFD having a non-UFD power series ring
Square of a torch ring
Trivial extension torch ring