|
$2$-adic integers: $\mathbb Z_2$ |
|
|
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$ |
|
|
$\mathbb C$: the field of complex numbers |
|
|
$\mathbb Q$: the field of rational numbers |
|
|
$\mathbb Q(x)$: rational functions over the rational numbers |
|
|
$\mathbb Q[[X]]$ |
|
|
$\mathbb Q[[x^2,x^3]]$ |
|
|
$\mathbb Q[\mathbb Q]$ |
|
|
$\mathbb Q[x,x^{-1}]$: Laurent polynomials |
|
|
$\mathbb Q[x,y,z]/(xz,yz)$ |
|
|
$\mathbb Q[x,y]/(x^2, xy)$ |
|
|
$\mathbb Q[x,y]/(x^2-y^3)$ |
|
|
$\mathbb Q[X,Y]_{(X,Y)}$ |
|
|
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$ |
|
|
$\mathbb Q[x]$ |
|
|
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$ |
|
|
$\mathbb Q[x_1, x_2,\ldots, x_n]$ |
|
|
$\mathbb R$: the field of real numbers |
|
|
$\mathbb R[[x]]$ |
|
|
$\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2-1)$ |
|
|
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$ |
|
|
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$ |
|
|
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions |
|
|
$\mathbb R[x]/(x^2)$ |
|
|
$\mathbb R[x_1, x_2,x_3,\ldots]$ |
|
|
$\mathbb Z$: 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, $k>1$ |
|
|
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$ |
|
|
$\mathbb Z[\sqrt{-5}]$ |
|
|
$\mathbb Z[i]$: the Gaussian integers |
|
|
$\mathbb Z[x]$ |
|
|
$\mathbb Z[X]/(X^2,4X, 8)$ |
|
|
$\mathbb Z[X]/(X^2,8)$ |
|
|
$\mathbb Z[x]/(x^2-1)$ |
|
|
$\mathbb Z[x_0, x_1,x_2,\ldots]$ |
|
|
$\mathbb Z_S$, where $S=((2)\cup(3))^c$ |
|
|
$\mathbb Z_{(2)}$ |
|
|
$\prod_{i=0}^\infty \mathbb Q$ |
|
|
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$ |
|
|
$\prod_{i=1}^\infty \mathbb Z/(2^i)$ |
|
|
$\prod_{i=1}^\infty F_2$ |
|
|
$\varinjlim \mathbb Q^{2^n}$ |
|
|
$\widehat{\mathbb Z}$: the profinite completion of the integers |
|
|
$^\ast \mathbb R$: the field of hyperreal numbers |
|
|
$C([0,1])$, the ring of continuous real-valued functions on the unit interval |
|
|
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$ |
|
|
$F_2[x,y]/(x,y)^2$ |
|
|
$F_p(x)$ |
|
|
$k[[x,y]]/(x^2,xy)$ |
|
|
10-adic numbers |
|
|
2-truncated Witt vectors over $\Bbb F_2((t))$ |
|
|
Akizuki's counterexample |
|
|
Algebraic closure of $F_2$ |
|
|
Algebraic integers |
|
|
Base ring for $R_{187}$ |
|
|
catenary, not universally catenary |
|
|
Clark's uniserial ring |
|
|
Cohn's Schreier domain that isn't GCD |
|
|
Countably infinite boolean ring |
|
|
Custom Krull dimension valuation ring |
|
|
DVR that is not N-2 |
|
|
Eventually constant sequences in $\mathbb Z$ |
|
|
Facchini's torch ring |
|
|
field of $2$-adic numbers |
|
|
Field of algebraic numbers |
|
|
Field of constructible numbers |
|
|
Finitely cogenerated, not semilocal ring |
|
|
Grams' atomic domain which doesn't satisfy ACCP |
|
|
Henselization of $\Bbb Z_{(2)}$ |
|
|
Hochster's connected, nondomain, locally-domain ring |
|
|
Interval monoid ring |
|
|
Kasch not semilocal ring |
|
|
Kerr's Goldie ring with non-Goldie matrix ring |
|
|
McGovern's commutative Zorn ring that isn't clean |
|
|
Mori but not Krull domain |
|
|
Nagata ring that not quasi-excellent |
|
|
Nagata's Noetherian infinite Krull dimension ring |
|
|
Nagata's normal ring that is not analytically normal |
|
|
Noetherian domain that is not N-1 |
|
|
Noetherian ring that is not Grothendieck and not Nagata |
|
|
non-$h$-local domain |
|
|
Osofsky's Type I ring |
|
|
Perfect non-Artinian ring |
|
|
Perfect ring that isn't semiprimary |
|
|
Principal ideal domain that is not Nagata |
|
|
Progression free polynomial ring |
|
|
Pseudo-Frobenius, not quasi-Frobenius ring |
|
|
Quasi-continuous ring that is not Ikeda-Nakayama |
|
|
reduced $I_0$ ring that is not exchange |
|
|
reduced exchange ring which is not semiregular |
|
|
ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$ |
|
|
Ring of holomorphic functions on $\mathbb C$ |
|
|
Ring of integer valued polynomials over the rationals |
|
|
Samuel's UFD having a non-UFD power series ring |
|
|
Square of a torch ring |
|
|
Trivial extension torch ring |
|