Akizuki's counterexample 

Algebraic closure of $F_2$ 

Algebraic integers 

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

Clark's uniserial ring 

Cohn's Schreier domain that isn't GCD 

Countably infinite boolean ring 

Custom Krull dimension valuation ring 

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

Field of algebraic numbers 

Field of constructible numbers 

field of $p$adic numbers 

Finitely cogenerated, not semilocal ring. 

$F_p(x)$ 

Grams' atomic domain which doesn't satisfy ACCP 

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

Hochster's connected, nondomain, locallydomain ring 

Interval monoid ring 

Kasch not semilocal ring 

$k[[x]]$ 

$k[x]$ 

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

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

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

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

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

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

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

$\mathbb C$: the field of complex numbers 

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

$\mathbb Q[\mathbb Q]$ 

$\mathbb Q$: 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 

$\mathbb Z/(n)$, $n$ 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 

Mori but not Krull domain 

Nagata's Noetherian infinite Krull dimension ring 

Nagata's normal ring that is not analytically normal 

Noetherian ring that is not Grothendieck and not Nagata 

$p$adic integers: $\mathbb Z_p$ 

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 

reduced exchange ring which is not semiregular 

reduced $I_0$ ring that is not exchange 

Ring of holomorphic functions on $\mathbb C$ 

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