$2$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 

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

$\mathbb Q[\mathbb Q]$ 

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

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

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

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

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

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

$\mathbb Q[x]$ 

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

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

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

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

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

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

$\mathbb Z[x]$ 

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

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

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

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

$\mathbb Z_S$, 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 realvalued functions on the unit interval 

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

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

$F_p(x)$ 

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

10adic numbers 

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

Akizuki's counterexample 

Algebraic closure of $F_2$ 

Algebraic integers 

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 N2 

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, locallydomain ring 

Interval monoid ring 

Kasch not semilocal ring 

Kerr's Goldie ring with nonGoldie matrix ring 

McGovern's commutative Zorn ring that isn't clean 

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 

Noetherian domain that is not N1 

Noetherian ring that is not Grothendieck and not Nagata 

non$h$local domain 

Osofsky's Type I ring 

Perfect nonArtinian ring 

Perfect ring that isn't semiprimary 

Progression free polynomial ring 

PseudoFrobenius, not quasiFrobenius ring 

Quasicontinuous ring that is not IkedaNakayama 

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 $\mathbb C$ 

Samuel's UFD having a nonUFD power series ring 

Square of a torch ring 

Trivial extension torch ring 
