View rings by dimension type

If you find a measurement for a ring is missing, please report it with the suggestion form!

Columns can be sorted by clicking the headers. The quantities are sorted as strings, not as cardinals.

Type: cardinality

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

Show commutative rings only

Name Dimension
$T_n(F_2)$ $2^{n(n+1)/2}$
$M_n(F_2)$ $2^{n^2}$
$\mathbb Q$: 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 $\aleph_0$
$\mathbb Z[x]$ $\aleph_0$
$\mathbb Z[x]/(x^2-1)$ $\aleph_0$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$ $\aleph_0$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$ $\aleph_0$
$\mathbb Z_{(2)}$ $\aleph_0$
$F_p(x)$ $\aleph_0$
$k[x,x^{-1};\sigma]$ $\aleph_0$
$k[x,y,z]/(xz,yz)$ $\aleph_0$
$k[x,y]/(x^2, xy)$ $\aleph_0$
$k[x,y]/(x^2-y^3)$ $\aleph_0$
$k[x,y]_{(x,y)}/(x^2-y^3)$ $\aleph_0$
$k[x;\sigma]/(x^2)$ (Artinian) $\aleph_0$
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$ $\aleph_0$
$M_n(\mathbb Q)$ $\aleph_0$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$ $\aleph_0$
Algebra of differential operators on the line (1st Weyl algebra) $\aleph_0$
Algebraic integers $\aleph_0$
Bass's right-not-left perfect ring $\aleph_0$
Bergman's example showing that "compressible" is not Morita invariant $\aleph_0$
Cohn's non-IBN domain $\aleph_0$
Countably infinite boolean ring $\aleph_0$
Cozzens' simple V-domain $\aleph_0$
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 $\aleph_0$
Hurwitz quaternions $\aleph_0$
Kasch not semilocal ring $\aleph_0$
Left-not-right Noetherian domain $\aleph_0$
Lipschitz quaternions $\aleph_0$
Nagata's Noetherian infinite Krull dimension ring $\aleph_0$
Non-symmetric $2$-primal 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 $\aleph_0$
Right-not-left ACC on annihilators triangular ring $\aleph_0$
Right-not-left Artinian triangular ring $\aleph_0$
Right-not-left coherent ring $\aleph_0$
Right-not-left Noetherian triangular ring $\aleph_0$
Right-not-left nonsingular ring $\aleph_0$
Shepherdson's domain that is not stably finite $\aleph_0$
Varadarajan's left-not-right coHopfian ring $\aleph_0$
$2$-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 $\mathfrak c$
$\mathbb R[x]/(x^2)$ $\mathfrak c$
$\mathbb R[x_1, x_2,x_3,\ldots]$ $\mathfrak c$
$\prod_{i=1}^\infty F_2$ $\mathfrak c$
Berberian's incompressible Baer ring $\mathfrak c$
Chase's left-not-right semihereditary ring $\mathfrak c$
Clark's uniserial ring $\mathfrak c$
field of $2$-adic numbers $\mathfrak c$
Full linear ring of a countable dimensional right vector space $\mathfrak c$
Interval monoid ring $\mathfrak c$
Local right-not-left Kasch ring $\mathfrak c$
Puninski's triangular serial ring $\mathfrak c$
Ring of holomorphic functions on $\mathbb C$ $\mathfrak c$
$\mathbb C$: the field of complex numbers $\mathfrak{c}$
$\mathbb H$: Hamilton's quaternions $\mathfrak{c}$
$\mathbb R$: 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, $k>1$ $p^k$
$T_n(F_q)$ $q^{n(n+1)/2}$
Basic ring of Nakayama's QF ring 16
$F_2[S_4]$ 16,777,216
$\mathbb Z/(2)$ 2
$F_2[\mathcal Q_8]$ 256
$\mathbb Z[X]/(X^2,4X, 8)$ 32
Osofsky's $32$ element ring 32
Right-not-left Kasch ring 32
Nakayama's quasi-Frobenius ring that isn't Frobenius 512
$\mathbb Z[X]/(X^2,8)$ 64
$F_2[x,y]/(x,y)^2$ 8
Reversible non-symmetric ring 8192
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$k[[x,y]]/(x^2,xy)$
$k[x;\sigma]/(x^2)$ (not right Artinian)
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
10-adic numbers
2-dimensional uniserial domain
Akizuki's counterexample
Algebraic closure of $F_2$
Bergman's exchange ring that isn't clean
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
Division algebra with no anti-automorphism
Division ring with an antihomomorphism but no involution
DVR that is not N-2
Eventually constant sequences in $\mathbb Z$
Facchini's torch ring
Faith-Menal counterexample
Goodearl's simple self-injective operator algebra
Goodearl's simple self-injective von Neumann regular ring
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
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
Left-not-right uniserial domain
Malcev's nonembeddable domain
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 normal ring that is not analytically normal
Nielsen's right UGP, not left UGP ring
Nielsen's semicommutative ring that isn't McCoy
Noetherian domain that is not N-1
Noetherian ring that is not Grothendieck and not Nagata
non-$h$-local domain
Non-Artinian simple ring
Nonlocal endomorphism ring of a uniserial module
O'Meara's infinite matrix algebra
Osofsky's Type I ring
Page's left-not-right FPF ring
Progression free polynomial ring
Pseudo-Frobenius, not quasi-Frobenius ring
Ram's Ore extension 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
Semicommutative $R$ such that $R[x]$ is not semicommutative
Simple, connected, Noetherian ring with zero divisors
Simple, non-Artinian, von Neumann regular ring
Small's right hereditary, not-left semihereditary ring
Square of a torch ring
Trivial extension torch ring
Šter's clean ring with non-clean corner rings
Šter's counterexample showing "clean" is not Morita invariant