$2$adic integers: $\mathbb Z_2$ 

$C([0,1])$, the ring of continuous realvalued functions on the unit interval 

$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3space 

$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$ 

$F_2[S_4]$ 

$F_2[\mathcal Q_8]$ 

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

$F_p(x)$ 

$M_n(F_2)$ 

$M_n(\mathbb Q)$ 

$RCFM_\omega(\mathbb Q)$ 

$T_\omega(\mathbb Q)$ 

$T_n(F_2)$ 

$T_n(F_q)$ 

$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$ 

$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$ 

$\mathbb C$: the field of complex numbers 

$\mathbb H$: Hamilton's quaternions 

$\mathbb Q$: the field of rational numbers 

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

$\mathbb Q+FM_\omega(\mathbb Q)$ 

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

$\mathbb Q[[x^2,x^3]]$ 

$\mathbb Q[\mathbb Q]$ 

$\mathbb Q[x,x^{1}]$: Laurent polynomials 

$\mathbb Q[x,y,z]/(xz,yz)$ 

$\mathbb Q[x,y]/(x^2, xy)$ 

$\mathbb Q[x,y]/(x^2y^3)$ 

$\mathbb Q[x,y]_{(x,y)}/(x^2y^3)$ 

$\mathbb Q[x]$ 

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

$\mathbb Q[x_1, x_2,\ldots, x_n]$ 

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

$\mathbb Q\langle x, y\rangle$ 

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

$\mathbb R$: the field of real numbers 

$\mathbb R[[x]]$ 

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

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

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

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

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

$\mathbb Z$: 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, $k>1$ 

$\mathbb Z[X]/(X^2,4X, 8)$ 

$\mathbb Z[X]/(X^2,8)$ 

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

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

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

$\mathbb Z[x]$ 

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

$\mathbb Z[x_0, x_1,x_2,\ldots]$ 

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

$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$ 

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

$\mathbb Z_{(2)}$ 

$\prod_{i=0}^\infty \mathbb Q$ 

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

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

$\prod_{i=1}^\infty \mathbb Z/(2^i)$ 

$\varinjlim T_{2^n}(\Bbb Q)$ 

$\varinjlim M_{2^n}(\mathbb Q)$ 

$\varinjlim \mathbb Q^{2^n}$ 

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

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

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

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

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

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

10adic numbers 

2dimensional uniserial domain 

2truncated Witt vectors over $\Bbb F_2((t))$ 

Akizuki's counterexample 

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 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 ring with IBN 

Bergman's ring without IBN 

Bergman's unitregular ring 

Camillo and Nielsen's McCoy ring 

Chase's leftnotright semihereditary ring 

Clark's uniserial ring 

Cohn's Schreier domain that isn't GCD 

Cohn's nonIBN domain 

Cohn's rightnotleft free ideal ring 

Countably infinite boolean ring 

Cozzens simple, left principal, right nonNoetherian domain 

Cozzens' simple Vdomain 

Custom Krull dimension valuation ring 

DVR that is not N2 

Division algebra with no antiautomorphism 

Division ring with an antihomomorphism but no involution 

Eventually constant sequences in $\mathbb Z$ 

Facchini's torch ring 

FaithMenal counterexample 

Field of algebraic numbers 

Field of constructible numbers 

Finitely cogenerated, not semilocal ring 

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 

Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$ 

Henselization of $\Bbb Z_{(2)}$ 

Hochster's connected, nondomain, locallydomain ring 

Hurwitz quaternions 

Interval monoid ring 

Kaplansky's rightnotleft hereditary ring 

Kasch not semilocal ring 

Kerr's Goldie ring with nonGoldie matrix ring 

Kolchin's simple Vdomain 

Leavitt path algebra of an infinite bouquet of circles 

Leftnotright Noetherian domain 

Leftnotright pseudoFrobenius ring 

Leftnotright uniserial domain 

Lipschitz quaternions 

Local rightnotleft Kasch ring 

Malcev's nonembeddable domain 

McGovern's commutative Zorn ring that isn't clean 

Michler & Villamayor's rightnotleft V ring 

Mori but not Krull domain 

Nagata ring that not quasiexcellent 

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 right UGP, not left UGP ring 

Nielsen's semicommutative ring that isn't McCoy 

Noetherian domain that is not N1 

Noetherian ring that is not Grothendieck and not Nagata 

NonArtinian simple ring 

Nonsymmetric $2$primal ring 

Nonlocal endomorphism ring of a uniserial module 

O'Meara's infinite matrix algebra 

Osofsky's $32$ element ring 

Osofsky's Type I ring 

Page's leftnotright FPF ring 

Perfect nonArtinian ring 

Perfect ring that isn't semiprimary 

Progression free polynomial ring 

PseudoFrobenius, not quasiFrobenius ring 

Puninski's triangular serial ring 

Quasicontinuous ring that is not IkedaNakayama 

Ram's Ore extension ring 

Rational quaternions 

Reversible nonsymmetric ring 

Rightnotleft ACC on annihilators triangular ring 

Rightnotleft Artinian triangular ring 

Rightnotleft Kasch ring 

Rightnotleft Noetherian triangular ring 

Rightnotleft coherent ring 

Rightnotleft nonsingular ring 

Rightnotleft simple injective ring 

Ring of holomorphic functions on $\mathbb C$ 

Samuel's UFD having a nonUFD power series ring 

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

Shepherdson's domain that is not stably finite 

Simple, Noetherian ring with zero divisors and trivial idempotents 

Simple, nonArtinian, von Neumann regular ring 

Small's right hereditary, notleft semihereditary ring 

Square of a torch ring 

Trivial extension torch ring 

Varadarajan's leftnotright coHopfian ring 

catenary, not universally catenary 

field of $2$adic numbers 

non$h$local domain 

reduced $I_0$ ring that is not exchange 

reduced exchange ring which is not semiregular 

ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$ 

Šter's clean ring with nonclean corner rings 

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