Property: almost maximal domain

Definition: $R$ is an integral domain, is $h$-local, and the localization at each maximal ideal is an almost maximal valuation ring. In Brandal, Willy. "Almost maximal integral domains and finitely generated modules." Transactions of the American Mathematical Society 183 (1973): 203-222. Theorem 2.9 this is shown to be equivalent to "almost maximal ring + domain")

	Name
	$\mathbb C$: the field of complex numbers
	$\mathbb Q$: the field of rational numbers
	$\mathbb Q(x)$: rational functions over the rational numbers
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb R$: the field of real numbers
	$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
	$\mathbb Z/(2)$
	$\mathbb Z/(p)$, $p$ an odd prime
	$\mathbb Z[x_0, x_1,x_2,\ldots]$
	$^\ast \mathbb R$: the field of hyperreal numbers
	$F_p(x)$
	Algebraic closure of $F_2$
	catenary, not universally catenary
	Cohn's Schreier domain that isn't GCD
	field of $2$-adic numbers
	Field of algebraic numbers
	Field of constructible numbers
	Grams' atomic domain which doesn't satisfy ACCP
	Kerr's Goldie ring with non-Goldie matrix ring
	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
	ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
	Ring of holomorphic functions on $\mathbb C$
	Samuel's UFD having a non-UFD power series ring
	$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
	$\mathbb R[x]/(x^2)$
	$\mathbb R[x_1, x_2,x_3,\ldots]$
	$\mathbb Z+x\mathbb Q[x]$
	$\mathbb Z/(n)$, $n$ divisible by two primes and a square
	$\mathbb Z/(n)$, $n$ squarefree and not prime.
	$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
	$\mathbb Z[x]$
	$\mathbb Z[X]/(X^2,4X, 8)$
	$\mathbb Z[X]/(X^2,8)$
	$\mathbb Z[x]/(x^2-1)$
	$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
	$\prod_{i=1}^\infty F_2$
	$\widehat{\mathbb Z}$: the profinite completion of the integers
	$C([0,1])$, 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 $0$
	$F_2[x,y]/(x,y)^2$
	$k[[x,y]]/(x^2,xy)$
	$k[x,y,z]/(xz,yz)$
	$k[x,y]/(x^2, xy)$
	$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	10-adic numbers
	Algebraic integers
	Clark's uniserial ring
	Countably infinite boolean ring
	Custom Krull dimension valuation ring
	Eventually constant sequences in $\mathbb Z$
	Facchini's torch ring
	Finitely cogenerated, not semilocal ring.
	Hochster's connected, nondomain, locally-domain ring
	Interval monoid ring
	Kasch not semilocal ring
	McGovern's commutative Zorn ring that isn't clean
	non-$h$-local domain
	Perfect non-Artinian ring
	Perfect ring that isn't semiprimary
	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
	Square of a torch ring
	Trivial extension torch ring
	$2$-adic integers: $\mathbb Z_2$
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb Q[x]$
	$\mathbb R[[x]]$
	$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
	$\mathbb Z$: the ring of integers
	$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
	$\mathbb Z[\sqrt{-5}]$
	$\mathbb Z[i]$: the Gaussian integers
	$\mathbb Z_S$, where $S=((2)\cup(3))^c$
	$\mathbb Z_{(2)}$
	$k[x,y]/(x^2-y^3)$
	$k[x,y]_{(x,y)}/(x^2-y^3)$
	Akizuki's counterexample
	DVR that is not N-2
	Henselization of $\Bbb Z_{(2)}$
	Noetherian domain that is not N-1
	Noetherian ring that is not Grothendieck and not Nagata
	Osofsky's Type I ring

Property: almost maximal domain

Reference(s):

Metaproperties: