Commutative ring properties

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

$R$ 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

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

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

$R$ 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")

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

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

a local ring whose completion is a normal ring

a local ring whose completion is reduced

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

$R$ 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.)

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

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 $n$.

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

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

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

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

A domain whose ideals are projective modules

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

A domain which has a Euclidean valuation

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

$R$ is quasi-excellent and universally catenary

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

A domain having a gcd for every pair of elements

$R$ 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 $R$

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

Noetherian and its formal fibers are geometrically regular

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 $R[x]$.)

the set of regular points of the spectrum contains a non-empty open subset

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

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

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

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.

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

$R$ is commutative and linearly compact

$R$ 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 $R$.

A domain satisfying the ACC on integral divisorial ideals

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

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"

Noetherian and universally Japanese

Ring whose localizations at primes all are normal domains

domain that is integrally closed in its field of fractions

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

A field over which every irreducible polynomial has distinct roots.

A domain whose finitely generated ideals are projective modules

The sum of two squares is a square

Every element is a square of another element

$R$ is Grothendieck and J-2

$Nil(R)=J(R)$

Noetherian ring whose localizations at primes are regular local rings

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

normal domain in which every element is primal

$R$ 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.)

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

All finitely generated algebras over $R$ are catenary

every finitely generated integral domain over $R$ is Japanese

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

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