Rings

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

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

$F_2[\mathcal Q_8]$

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

$F_p(x)$

$k\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra

$k[[x]]$

$k[x]$

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

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

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

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

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

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

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

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

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

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

$\mathbb C$: the field of complex numbers

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

$\mathbb H$: Hamilton's quaternions

$\mathbb Q$: the field of rational numbers

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

$\mathbb Q[\mathbb Q]$

$\mathbb Q(x)$: 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 $I=(x^2+y^2-1)$

$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions

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

$\mathbb Z$: the ring of integers

$\mathbb Z/(2)$

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

$\mathbb Z[i]$: 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.

$\mathbb Z_{(p)}$

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

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

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

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

$\mathbb Z[x]$

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

$\mathbb Z[x]/(x^2-1)$

$M_n(F_q)$

$M_n(k)$

$p$-adic integers: $\mathbb Z_p$

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

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

$T_2(F_2)$

$T_n(F_q)$

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

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

Akizuki's counterexample

Algebraic closure of $F_2$

Algebraic integers

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

Bass's right-not-left 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 non-unit-regular subring

Bergman's primitive finite uniform dimension ring

Bergman's right-not-left primitive ring

Bergman's unit-regular ring

Chase's left-not-right semihereditary ring

Clark's uniserial ring

Cohn's non-IBN domain

Cohn's right-not-left free ideal ring

Cohn's Schreier domain that isn't GCD

Countably infinite boolean ring

Cozzens simple, left principal, right non-Noetherian domain

Cozzens' simple V-domain

Custom Krull dimension valuation ring

Faith-Menal counterexample

field of $p$-adic numbers

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 self-injective operator algebra

Goodearl's simple self-injective von Neumann regular ring

Grams' atomic domain which doesn't satisfy ACCP

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

Hochster's connected, nondomain, locally-domain ring

Hurwitz quaternions

Interval monoid ring

Kasch not semilocal ring

Kolchin's simple V-domain

Leavitt path algebra of an infinite bouquet of circles

Left-not-right Noetherian domain

Left-not-right pseudo-Frobenius ring

Left-not-right valuation domain

Lipschitz quaternions

Local right-not-left Kasch ring

Malcev's nonembeddable domain

McGovern's commutative Zorn ring that isn't clean

Michler & Villemayor's right-not-left V ring

Mori but not Krull domain

Nagata's Noetherian infinite Krull dimension ring

Nagata's normal ring that is not analytically normal

Nakayama's quasi-Frobenius ring that isn't Frobenius

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-Artinian simple ring

Non-symmetric $2$-primal ring

O'Meara's infinite matrix algebra

Osofsky's $32$ element ring

Page's left-not-right FPF ring

Perfect non-Artinian ring

Perfect ring that isn't semiprimary

Pseudo-Frobenius, not quasi-Frobenius ring

Puninski's triangular serial ring

reduced $I_0$ ring that is not exchange

reduced exchange ring which is not semiregular

Reversible non-symmetric ring

Right-not-left ACC on annihilators triangular ring

Right-not-left Artinian triangular ring

Right-not-left coherent ring

Right-not-left Kasch ring

Right-not-left Noetherian triangular ring

Right-not-left nonsingular ring

Right-not-left simple injective ring

Ring of holomorphic functions on $\mathbb C$

Samuel's UFD having a non-UFD power series ring

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, non-Artinian, von Neumann regular ring

Šter's clean ring with non-clean corner rings

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

Varadarajan's left-not-right coHopfian ring