This is the list of ring properties defined only for commutative rings.
Name  Definition  % Complete 

$h$local domain 
A commutative domain $R$ for which every nontrivial ideal is contained in finitely many maximal ideals, and every nonzero prime is contained in a unique maximal ideal 

?ring 
$R$ is 1) arithmetical; 2) has a unique, nonzero, minimal prime ideal $P$ that is uniserial as an $R$ module; 3) $R/P$ is an hlocal domain; 4)$R$ is not local 

algebraically closed field 
Every nonconstant polynomial over the field has a root in the field 

almost Dedekind domain 
A commutative integral domain whose localizations at maximal ideals are all discrete valuation rings, or a field 

almost maximal domain 
$R$ 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): 203222. Theorem 2.9 this is shown to be equivalent to "almost maximal ring + domain") 

almost maximal ring 
$R/I$ is a maximal ring for every nonzero ideal $I$ 

almost maximal valuation ring 
$R$ is a commutative uniserial ring such that for every nonzero ideal $I$, the quotient $R/I$ is a maximal valuation ring. 

analytically normal 
a local ring whose completion is a normal ring 

analytically unramified 
a local ring whose completion is reduced 

Archimedean field 
an ordered field such that for every $x$ there exists an integer $n>x$. 

arithmetical 
$R$ is commutative and the localization at each maximal ideal is a uniserial ring. (This is known to be equivalent to $R$ being a commutative distributive ring.) 

atomic domain 
A domain in which nonzero nonunits can be written as a finite product of irreducible elements. 

catenary 
A commutative ring is called catenary if there exists an $n$ such that any chain of prime ideals can be refined to a maximal chain of prime ideals with length $n$. 

characteristic 0 field 
The sum of any positive number of 1's is always nonzero. 

CohenMacaulay 
Noetherian ring whose localizations at primes all have depth equal to their Krull dimension 

complete discrete valuation ring 
$R$ is a valuation ring and it is complete with respect to the metric furnished by its valuation 

complete local 
$R$ is a commutative local ring in which the canonical map into the completion at the maximal ideal is an isomorphism. 

Dedekind domain 
A domain whose ideals are projective modules 

discrete valuation ring 
$R$ is a valuation ring with value group isomorphic to $(\mathbb Z,+)$ 

Euclidean domain 
A domain which has a Euclidean valuation 

Euclidean field 
An ordered field for which every positive element is the square of another element 

excellent 
$R$ is quasiexcellent and universally catenary 

FGC 
"Finitely Generated are Cyclic": A commutative ring for which every finitely generated module is a direct sum of cyclic modules. 

GCD domain 
A domain having a gcd for every pair of elements 

Goldman domain 
$R$ is a commutative domain with field of fractions $K$, and $K=R[u^{1}]$ for some nonzero $u\in R$. Equivalently: $K$ is finitely generated as a ring over $R$; Equivalently: There is a nonzero $u\in R$ contained in all nonzero prime ideals of $R$ 

Gorenstein 
Noetherian ring whose localizations at primes all are Noetherian and have finite injective dimension 

Grothendieck 
Noetherian and its formal fibers are geometrically regular 

Henselian local 
A commutative local ring in which Hensel's Lemma holds. (For any monic polynomial $p$ in $R[x]$, all factorizations in $(R/M)[x]$ into a product of coprime monic polynomials lift to factorizations in $R[x]$.) 

J0 
the set of regular points of the spectrum contains a nonempty open subset 

J1 
the set of regular points of the spectrum is an open subset 

J2 
for every finitely generated $R$algebra $S$, the singular points of $Spec(S)$ form a closed subset. 

Jacobson 
Prime ideals are intersections of maximal ideals. Also known as: Hilbert rings 

Krull domain 
Three conditions must hold: 1) localizations at height 1 primes are all discrete valuation rings; 2) $R$ is the intersection of those valuation rings in its field of fractions; 3) Each nonzero element of $R$ is contained in only finitely many height 1 primes. 

local complete intersection 
Noetherian local ring whose completion is the quotient of a regular local ring by an ideal generated by a regular sequence 

maximal ring 
$R$ is commutative and linearly compact 

maximal valuation ring 
$R$ is a commutative uniserial ring such that every system of congruences $x \equiv r_{\lambda } \mbox{mod} {\mathfrak {b}}_{\lambda } (\lambda \in \Lambda )$ which is pairwise soluable has a simultaneous solution in $R$. 

Mori domain 
A domain satisfying the ACC on integral divisorial ideals 

N1 
A domain $R$ whose integral closure in its quotient field is a finitely generated $R$ module. 

N2 
A domain $R$ such that for every finite extension $L$ of its quotient field $K$, the integral closure of $R$ in $L$ is a finitely generated Amodule. Also known as "Japanese rings" 

Nagata 
Noetherian and universally Japanese 

normal 
Ring whose localizations at primes all are normal domains 

normal domain 
domain that is integrally closed in its field of fractions 

ordered field 
the field is totally ordered with an order compatible with the ring operations 

perfect field 
A field over which every irreducible polynomial has distinct roots. 

Prufer domain 
A domain whose finitely generated ideals are projective modules 

Pythagorean field 
The sum of two squares is a square 

quadratically closed field 
Every element is a square of another element 

quasiexcellent 
$R$ is Grothendieck and J2 

radnil 
$Nil(R)=J(R)$ 

regular 
Noetherian ring whose localizations at primes are regular local rings 

regular local 
Noetherian local ring where the minimal number of generators for the maximal ideal is equal to its Krull dimension 

Schreier domain 
normal domain in which every element is primal 

torch 
$R$ is a commutative ring meeting these conditions: 1) $R$ is not local; 2) $R$ has a unique minimal prime $P$ which is a nonzero uniserial module; 3) $R/P$ is an $h$local domain; 4) $R$ is Bezout; 5) The localization $R_M$ is an almost maximal valuation ring for every maximal ideal $M$. (Stronger than the definition given by Vámos.) 

unique factorization domain 
A domain in which every nonzero nonunit is a product of irreducible elements, unique up to equivalence 

universally catenary 
All finitely generated algebras over $R$ are catenary 

universally Japanese 
every finitely generated integral domain over $R$ is Japanese 

valuation domain 
A commutative integral domain whose ideals are linearly ordered. (In some places also called a "valuation ring") 

valuation ring 
A commutative ring whose ideals are linearly ordered (not necessarily a domain.) 