Akizuki's counterexample 

Algebraic closure of $F_2$ 

Algebraic integers 

Algebra of differential operators on the line (1st Weyl algebra) 

$^\ast \mathbb R$: the field of hyperreal numbers 

Bass's rightnotleft perfect ring 

Berberian's incompressible Baer ring 

Bergman's example showing that "compressible" is not Morita invariant 

Bergman's exchange ring that isn't clean 

Bergman's nonunitregular subring 

Bergman's primitive finite uniform dimension ring 

Bergman's rightnotleft primitive ring 

Bergman's unitregular ring 

Chase's leftnotright semihereditary ring 

Clark's uniserial ring 

Cohn's nonIBN domain 

Cohn's rightnotleft free ideal ring 

Cohn's Schreier domain that isn't GCD 

Countably infinite boolean ring 

Cozzens simple, left principal, right nonNoetherian domain 

Cozzens' simple Vdomain 

Custom Krull dimension valuation ring 

$F_2[\mathcal Q_8]$ 

$F_2[x,y]/(x,y)^2$ 

FaithMenal counterexample 

Field of algebraic numbers 

Field of constructible numbers 

field of $p$adic numbers 

Finitely cogenerated, not semilocal ring. 

$F_p(x)$ 

Full linear ring of a countable dimensional right vector space 

Goodearl's simple selfinjective operator algebra 

Goodearl's simple selfinjective von Neumann regular ring 

Grams' atomic domain which doesn't satisfy ACCP 

Henselization of $\Bbb Z_{(p)}$ 

Hochster's connected, nondomain, locallydomain ring 

Hurwitz quaternions 

Interval monoid ring 

Kasch not semilocal ring 

$k\langle x,y \rangle/(xy1)$: the ToeplitzJacobson algebra 

Kolchin's simple Vdomain 

$k[[x]]$ 

$k[x]$ 

$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$ 

$k[x_1, x_2,\ldots, x_n]$ 

$k[[x^2,x^3]]$ 

$k[x;\sigma]/(x^2)$ (Artinian) 

$k[x;\sigma]/(x^2)$ (not right Artinian) 

$k[x,x^{1};\sigma]$ 

$k[x,y]/(x^2, xy)$ 

$k[x,y]/(x^2y^3)$ 

$k[x,y]_{(x,y)}/(x^2y^3)$ 

$k[x,y,z]/(xz,yz)$ 

Leftnotright Noetherian domain 

Leftnotright pseudoFrobenius ring 

Leftnotright valuation domain 

Lipschitz quaternions 

Local rightnotleft Kasch ring 

Malcev's nonembeddable domain 

$\mathbb C$: the field of complex numbers 

$\mathbb F_2[x_1, x_2, x_3\ldots ]/(\{x_i^3\mid i\in \mathbb N\}\cup\{x_ix_ji\neq j\}\cup\{x_i^2m\mid i\in \mathbb N\})$ 

$\mathbb H$: Hamilton's quaternions 

$\mathbb Q\langle a,b\rangle/(a^2)$ 

$\mathbb Q[\mathbb Q]$ 

$\mathbb Q$: the field of rational numbers 

$\mathbb Q(x)$: rational functions over the rational numbers 

$\mathbb Q[X,Y]_{(X,Y)}$ 

$\mathbb R$: the field of real numbers 

$\mathbb R[x_1, x_2,x_3,\ldots]$ 

$\mathbb R[x]/(x^2)$ 

$\mathbb R[x,y]$ completed $I$adically with $I=(x^2+y^21)$ 

$\mathbb R[x,y]/(x^2+y^21)$: ring of trigonometric functions 

$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2xy)$ 

$\mathbb Z/(2)$ 

$\mathbb Z[\frac{1+\sqrt{19}}{2}]$ 

$\mathbb Z[i]$: the Gaussian integers 

$\mathbb Z\langle x,y\rangle/(y^2, yx)$ 

$\mathbb Z/(n)$, $n$ divisible by two primes and a square 

$\mathbb Z/(n)$, $n$ squarefree and not prime. 

$\mathbb Z_{(p)}$ 

$\mathbb Z/(p^k)$, $p$ a prime, $k>1$ 

$\mathbb Z/(p)$, $p$ an odd prime 

$\mathbb Z[\sqrt{5}]$ 

$\mathbb Z_S$, where $S=((2)\cup(3))^c$ 

$\mathbb Z$: the ring of integers 

$\mathbb Z[x]$ 

$\mathbb Z+x\mathbb Q[x]$ 

$\mathbb Z[x]/(x^21)$ 

McGovern's commutative Zorn ring that isn't clean 

Michler & Villemayor's rightnotleft V ring 

$M_n(F_q)$ 

$M_n(k)$ 

Mori but not Krull domain 

Nagata's Noetherian infinite Krull dimension ring 

Nagata's normal ring that is not analytically normal 

Nakayama's quasiFrobenius ring that isn't Frobenius 

Nielsen's semicommutative ring that isn't McCoy 

Noetherian ring that is not Grothendieck and not Nagata 

NonArtinian simple ring 

Nonsymmetric $2$primal ring 

O'Meara's infinite matrix algebra 

Osofsky's $32$ element ring 

$p$adic integers: $\mathbb Z_p$ 

Page's leftnotright FPF ring 

Perfect nonArtinian ring 

Perfect ring that isn't semiprimary 

$\prod_{i=1}^\infty F_2$ 

$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$ 

PseudoFrobenius, not quasiFrobenius ring 

Puninski's triangular serial ring 

reduced exchange ring which is not semiregular 

reduced $I_0$ ring that is not exchange 

Reversible nonsymmetric ring 

Rightnotleft ACC on annihilators triangular ring 

Rightnotleft Artinian triangular ring 

Rightnotleft coherent ring 

Rightnotleft Kasch ring 

Rightnotleft Noetherian triangular ring 

Rightnotleft nonsingular ring 

Rightnotleft simple injective ring 

Ring of holomorphic functions on $\mathbb C$ 

Semicommutative $R$ such that $R[x]$ is not semicommutative 

Shepherdson's domain that is not stably finite 

Simple, connected, Noetherian ring with zero divisors 

Simple, nonArtinian, von Neumann regular ring 

Šter's clean ring with nonclean corner rings 

Šter's counterexample showing "clean" is not Morita invariant 

$T_2(F_2)$ 

$T_n(F_q)$ 

$T_n(k)$: the upper triangular matrix ring over a field 

Varadarajan's leftnotright coHopfian ring 

$\widehat{\mathbb Z}$: the profinite completion of the integers 
