|
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, locally-domain ring |
|
|
Interval monoid ring |
|
|
Kasch not semilocal ring |
|
|
$k[[x]]$ |
|
|
$k[x]$ |
|
|
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$ |
|
|
$k[x_1, x_2,\ldots, x_n]$ |
|
|
$k[[x^2,x^3]]$ |
|
|
$k[x,y]/(x^2, xy)$ |
|
|
$k[x,y]/(x^2-y^3)$ |
|
|
$k[x,y]_{(x,y)}/(x^2-y^3)$ |
|
|
$k[x,y,z]/(xz,yz)$ |
|
|
$\mathbb C$: the field of complex numbers |
|
|
$\mathbb F_2[x_1, x_2, x_3\ldots ]/(\{x_i^3\mid i\in \mathbb N\}\cup\{x_ix_j|i\neq j\}\cup\{x_i^2-m\mid i\in \mathbb N\})$ |
|
|
$\mathbb Q[\mathbb Q]$ |
|
|
$\mathbb Q$: 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 $I=(x^2+y^2-1)$ |
|
|
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions |
|
|
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$ |
|
|
$\mathbb Z/(2)$ |
|
|
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$ |
|
|
$\mathbb Z[i]$: 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 |
|
|
$\mathbb Z[x]$ |
|
|
$\mathbb Z+x\mathbb Q[x]$ |
|
|
$\mathbb Z[x]/(x^2-1)$ |
|
|
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 |
|
|
$p$-adic integers: $\mathbb Z_p$ |
|
|
Perfect non-Artinian ring |
|
|
Perfect ring that isn't semiprimary |
|
|
$\prod_{i=1}^\infty F_2$ |
|
|
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$ |
|
|
Pseudo-Frobenius, not quasi-Frobenius 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 |
|