Rings

Name % Complete
Akizuki's counterexample
82%
Algebraic closure of $F_2$
100%
Algebraic integers
98%
Algebra of differential operators on the line (1st Weyl algebra)
70%
$^\ast \mathbb R$: the field of hyperreal numbers
100%
Bass's right-not-left perfect ring
62%
Berberian's incompressible Baer ring
79%
Bergman's example showing that "compressible" is not Morita invariant
45%
Bergman's exchange ring that isn't clean
70%
Bergman's non-unit-regular subring
68%
Bergman's primitive finite uniform dimension ring
64%
Bergman's right-not-left primitive ring
35%
Bergman's unit-regular ring
67%
Chase's left-not-right semihereditary ring
59%
Clark's uniserial ring
98%
Cohn's non-IBN domain
78%
Cohn's right-not-left free ideal ring
66%
Cohn's Schreier domain that isn't GCD
79%
Countably infinite boolean ring
89%
Cozzens simple, left principal, right non-Noetherian domain
82%
Cozzens' simple V-domain
91%
Custom Krull dimension valuation ring
99%
$F_2[\mathcal Q_8]$
99%
$F_2[x,y]/(x,y)^2$
98%
Faith-Menal counterexample
36%
Field of algebraic numbers
100%
Field of constructible numbers
100%
field of $p$-adic numbers
100%
Finitely cogenerated, not semilocal ring.
85%
$F_p(x)$
100%
Full linear ring of a countable dimensional right vector space
87%
Goodearl's simple self-injective operator algebra
87%
Goodearl's simple self-injective von Neumann regular ring
87%
Grams' atomic domain which doesn't satisfy ACCP
79%
Henselization of $\Bbb Z_{(p)}$
79%
Hochster's connected, nondomain, locally-domain ring
49%
Hurwitz quaternions
71%
Interval monoid ring
99%
Kasch not semilocal ring
77%
$k\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
61%
Kolchin's simple V-domain
90%
$k[[x]]$
99%
$k[x]$
94%
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
69%
$k[x_1, x_2,\ldots, x_n]$
95%
$k[[x^2,x^3]]$
97%
$k[x;\sigma]/(x^2)$ (Artinian)
95%
$k[x;\sigma]/(x^2)$ (not right Artinian)
94%
$k[x,x^{-1};\sigma]$
66%
$k[x,y]/(x^2, xy)$
76%
$k[x,y]/(x^2-y^3)$
95%
$k[x,y]_{(x,y)}/(x^2-y^3)$
94%
$k[x,y,z]/(xz,yz)$
93%
Left-not-right Noetherian domain
72%
Left-not-right pseudo-Frobenius ring
56.0%
Left-not-right valuation domain
93%
Lipschitz quaternions
71%
Local right-not-left Kasch ring
72%
Malcev's nonembeddable domain
68%
$\mathbb C$: the field of complex numbers
100%
$\mathbb F_2[x_1, x_2, x_3\ldots ]/(\{x_i^3\mid i\in \mathbb N\}\cup\{x_ix_j|i\neq j\}\cup\{x_i^2-m\mid i\in \mathbb N\})$
97%
$\mathbb H$: Hamilton's quaternions
100%
$\mathbb Q\langle a,b\rangle/(a^2)$
29.0%
$\mathbb Q[\mathbb Q]$
75%
$\mathbb Q$: the field of rational numbers
100%
$\mathbb Q(x)$: rational functions over the rational numbers
100%
$\mathbb Q[X,Y]_{(X,Y)}$
97%
$\mathbb R$: the field of real numbers
100%
$\mathbb R[x_1, x_2,x_3,\ldots]$
89%
$\mathbb R[x]/(x^2)$
100%
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
88%
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
93%
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
98%
$\mathbb Z/(2)$
100%
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
94%
$\mathbb Z[i]$: the Gaussian integers
94%
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
62%
$\mathbb Z/(n)$, $n$ divisible by two primes and a square
100%
$\mathbb Z/(n)$, $n$ squarefree and not prime.
100%
$\mathbb Z_{(p)}$
99%
$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
100%
$\mathbb Z/(p)$, $p$ an odd prime
100%
$\mathbb Z[\sqrt{-5}]$
93%
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
99%
$\mathbb Z$: the ring of integers
100%
$\mathbb Z[x]$
96%
$\mathbb Z+x\mathbb Q[x]$
80%
$\mathbb Z[x]/(x^2-1)$
76%
McGovern's commutative Zorn ring that isn't clean
65%
Michler & Villemayor's right-not-left V ring
64%
$M_n(F_q)$
99%
$M_n(k)$
98%
Mori but not Krull domain
71%
Nagata's Noetherian infinite Krull dimension ring
90%
Nagata's normal ring that is not analytically normal
68%
Nakayama's quasi-Frobenius ring that isn't Frobenius
84%
Nielsen's semicommutative ring that isn't McCoy
29.0%
Non-Artinian simple ring
39%
Non-symmetric $2$-primal ring
71%
O'Meara's infinite matrix algebra
42%
Osofsky's $32$ element ring
89%
$p$-adic integers: $\mathbb Z_p$
99%
Page's left-not-right FPF ring
60%
Perfect non-Artinian ring
93%
Perfect ring that isn't semiprimary
80%
$\prod_{i=1}^\infty F_2$
99%
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
61%
Pseudo-Frobenius, not quasi-Frobenius ring
88%
Puninski's triangular serial ring
48%
reduced exchange ring which is not semiregular
67%
reduced $I_0$ ring that is not exchange
66%
Reversible non-symmetric ring
72%
Right-not-left ACC on annihilators triangular ring
35%
Right-not-left Artinian triangular ring
71%
Right-not-left coherent ring
87%
Right-not-left Kasch ring
77%
Right-not-left Noetherian triangular ring
85%
Right-not-left nonsingular ring
85%
Right-not-left simple injective ring
54%
Ring of holomorphic functions on $\mathbb C$
98%
Semicommutative $R$ such that $R[x]$ is not semicommutative
26%
Shepherdson's domain that is not stably finite
77%
Simple, connected, Noetherian ring with zero divisors
73%
Simple, non-Artinian, von Neumann regular ring
84%
Šter's clean ring with non-clean corner rings
15%
Šter's counterexample showing "clean" is not Morita invariant
20%
$T_2(F_2)$
91%
$T_n(F_q)$
91%
$T_n(k)$: the upper triangular matrix ring over a field
90%
Varadarajan's left-not-right coHopfian ring
72%
$\widehat{\mathbb Z}$: the profinite completion of the integers
54%