# Commutative ring properties

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
89%
?-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 h-local domain; 4)$R$ is not local
98%
algebraically closed field
Every nonconstant polynomial over the field has a root in the field
100%
almost Dedekind domain
A commutative integral domain whose localizations at maximal ideals are all discrete valuation rings, or a field
90%
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): 203-222. Theorem 2.9 this is shown to be equivalent to "almost maximal ring + domain")
74%
almost maximal ring
$R/I$ is a maximal ring for every nonzero ideal $I$
45%
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.
94%
analytically normal
a local ring whose completion is a normal ring
78%
analytically unramified
a local ring whose completion is reduced
70%
Archimedean field
an ordered field such that for every $x$ there exists an integer $n>x$.
100%
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.)
76%
atomic domain
A domain in which nonzero nonunits can be written as a finite product of irreducible elements.
96%
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 $n$.
60%
characteristic 0 field
The sum of any positive number of 1's is always nonzero.
100%
Cohen-Macaulay
Noetherian ring whose localizations at primes all have depth equal to their Krull dimension
86%
complete discrete valuation ring
$R$ is a valuation ring and it is complete with respect to the metric furnished by its valuation
98%
complete local
$R$ is a commutative local ring in which the canonical map into the completion at the maximal ideal is an isomorphism.
88%
Dedekind domain
A domain whose ideals are projective modules
98%
discrete valuation ring
$R$ is a valuation ring with value group isomorphic to $(\mathbb Z,+)$
99%
Euclidean domain
A domain which has a Euclidean valuation
93%
Euclidean field
An ordered field for which every positive element is the square of another element
100%
excellent
$R$ is quasi-excellent and universally catenary
81%
FGC
"Finitely Generated are Cyclic": A commutative ring for which every finitely generated module is a direct sum of cyclic modules.
79%
GCD domain
A domain having a gcd for every pair of elements
95%
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 $R$
79%
Gorenstein
Noetherian ring whose localizations at primes all are Noetherian and have finite injective dimension
85%
Grothendieck
Noetherian and its formal fibers are geometrically regular
77%
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 $R[x]$.)
79%
J-0
the set of regular points of the spectrum contains a non-empty open subset
9%
J-1
the set of regular points of the spectrum is an open subset
36%
J-2
for every finitely generated $R$-algebra $S$, the singular points of $Spec(S)$ form a closed subset.
36%
Jacobson
Prime ideals are intersections of maximal ideals. Also known as: Hilbert rings
74%
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.
93%
local complete intersection
Noetherian local ring whose completion is the quotient of a regular local ring by an ideal generated by a regular sequence
86%
maximal ring
$R$ is commutative and linearly compact
84%
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 $R$.
95%
Mori domain
A domain satisfying the ACC on integral divisorial ideals
88%
N-1
A domain $R$ whose integral closure in its quotient field is a finitely generated $R$ module.
94%
N-2
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 A-module. Also known as "Japanese rings"
79%
Nagata
Noetherian and universally Japanese
89%
normal
Ring whose localizations at primes all are normal domains
81%
normal domain
domain that is integrally closed in its field of fractions
94%
ordered field
the field is totally ordered with an order compatible with the ring operations
100%
perfect field
A field over which every irreducible polynomial has distinct roots.
100%
Prufer domain
A domain whose finitely generated ideals are projective modules
95%
Pythagorean field
The sum of two squares is a square
99%
quadratically closed field
Every element is a square of another element
100%
quasi-excellent
$R$ is Grothendieck and J-2
81%
rad-nil
$Nil(R)=J(R)$
81%
regular
Noetherian ring whose localizations at primes are regular local rings
90%
regular local
Noetherian local ring where the minimal number of generators for the maximal ideal is equal to its Krull dimension
98%
Schreier domain
normal domain in which every element is primal
95%
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.)
99%
unique factorization domain
A domain in which every nonzero nonunit is a product of irreducible elements, unique up to equivalence
94%
universally catenary
All finitely generated algebras over $R$ are catenary
47%
universally Japanese
every finitely generated integral domain over $R$ is Japanese
50%
valuation domain
A commutative integral domain whose ideals are linearly ordered. (In some places also called a "valuation ring")
98%
valuation ring
A commutative ring whose ideals are linearly ordered (not necessarily a domain.)
94%