Ring properties

For all $a$ in $R$ there exists an $x\in R$ and a natural number $n$ such that $(a^n)x(a^n)=a^n$

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

Every right ideal not contained in $J(R)$ contains a nonzero idempotent. Also known as: semipotent.

All minimal prime ideals are completely prime

$R$ 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

All idempotents are central

Ascending chain condition on right annihilators=Descending chain condition on left annihilators

Ascending chain condition on principal right ideals

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

There exists an anti-isomorphism of $R$ into itself.

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.)

$R$ is called Armendariz if whenever $(\sum a_ix^i)(\sum b_jx^j)=0\in R[x]$, then $a_ib_j=0$ for all combinations of $i,j$.

Right Artinian (resp. left Artinian) Descending chain condition on right (resp. left) ideals.

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

For any subset $X$ of $R$, the right annihilator of $X$ is a summand of $R$

Finitely generated right ideals are cyclic

Right Bezout domain

For every $x\in R$, $x^2=x$.

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.

Every element is the sum of a unit and an idempotent

The ring cogenerates its category of right modules

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

Every finitely generated right ideal is finitely presented

For every $x$ with right annihilator zero, $x$ is a unit

$xy=yx$ for all $x$ and $y$ in the ring

$R$ 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.

Given any idempotent $e\in R$, we have $Z(eRe)=eZ(R)e$. ($Z(-)$ denotes the center of the ring.) (See Berberian's Baer and Baer * Rings, definition 3.29)

$R$ is right CS, and any right ideal isomorphic to a direct summand of $R$ is itself a summand.

The underlying set of the ring is countable.

"Complements are Summands": Every nonzero right ideal is essential in a direct summand of $R_R$.

Descending chain condition on right annihilators=Ascending chain condition on left annihilators

A domain whose ideals are projective modules

For all $x,y\in R$, $xy=1$ implies $yx=1$. Also known as: directly finite, von Neumann finite

The only central idempotents are $0$ and $1$. Also known as: connected (especially for commutative rings)

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

Lattice of right ideals is a distributive lattice

All nonzero elements are units. Also known as: skew-fields, sfields

No nonzero zero divisors. Also known as: entire rings (mainly for commutative domains)

For every right ideal $B$, $r.ann(l.ann(B))=B$

All right ideals are two-sided ideals

Essential right socle

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

If $a+b=1$, there exists $r$ and $s$ such that $ar$ and $bs$ are idempotents and $ar+bs=1$

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

If $T$ is a right ideal of $R$, and $f:T\to R$ such that f(T) is finitely generated, then $f$ can be extended to $R\to R$

A commutative division ring

$R$ has only finitely many elements

$R$ has finite uniform dimension as a right $R$ module

The ring is finitely cogenerated as a right module

Finitely generated right socle

Every finitely generated faithful right module is a generator for the category of $R$ modules

All right ideals are free modules and of unique rank

$R$ is quasi-Frobenius and the right socle is isomorphic to $R/J(R)$

all proper ideals are prime

all proper ideals are semiprime

A domain having a gcd for every pair of elements

$R$ has finite right uniform dimension and ACC on right annihilators

$R$ 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 $R[x]$.)

All right ideals are projective

(Invariant Basis Number) If $R^n$ is isomorphic to $R^m$, then $n=m$

"Internal Cancellation": If $A\oplus B=R$ and $A'\oplus B'=R$ are two decompositions of $R$ into right ideals, and if $A\cong A'$, then also $B\cong B'$. (This condition turns out to be left-right symmetric.)

The left annihilator of the intersection of two right ideals, is the sum of their left annihilators

There exists an anti-automorphism $\tau:R\to R$ such that $\tau^2$ is the identity map on $R$.

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

Every simple right module is isomorphic to a minimal right ideal of $R$

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.

Idempotents of $R/J(R)$ lift to idempotents in R

A ring is right linearly compact if $R_R$ is linearly compact as a module. That is, every finitely-solvable system of congruences using right ideals is solvable.

$R/J(R)$ is a division ring

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

(right max ring) A ring $R$ is called a right max ring if every nonzero right $R$ module has a maximal submodule. Also known as: "right $B$-ring" by Faith; "ring satisfying condition H on the right" by several French authors.

$R$ 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 ring $R$ is called right McCoy if when $f,g\in R[x]$ satisfy $fg=0$, then there exists a nonzero $r\in R$ such that $fr=0$.

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

"Nilpotents form an Ideal": The set of nilpotent elements in $R$ forms an ideal.

Has nil Jacobson radical

Has nilpotent Jacobson radical

Ascending chain condition on right ideals

The right singular ideal is zero

Nonzero right socle

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

Domain satisfying right Ore condition

Satisfies right Ore conditions

families of orthogonal idempotents are all finite

(right PCI ring) "Proper Cyclics are Injective": The proper right cyclic modules of $R$ (defined as the quotients not isomorphic to $R_R$) are injective $R$-modules.

$R$ is semilocal and $J(R)$ is right T-nilpotent

A field over which every irreducible polynomial has distinct roots.

For every element $x\in R$, there exists a natural number $n_x> 1$ such that $x^{n_x}=x$.

There exists an element of $\mathbb Z\langle x_1,\ldots x_n\rangle$ for which any set of $n$ ring elements satisfies the polynomials

The ring is $I_0$ and lift/rad

$R$ semiprimary and $R/J(R)$ simple

The product of nonzero right ideals is nonzero

There exists a faithful simple right $R$ module

Domain which is a principal right ideal ring

All right ideals are cyclic

homomorphisms from principal right ideals of the ring into the ring extend to endomorphisms of the ring

A domain whose finitely generated ideals are projective modules

$R$ is right self-injective and finitely cogenerated as a right module

The sum of two squares is a square

Every element is a square of another element

$R$ is right CS, and if $e,f$ are idempotents with $eR\cap fR=\{0\}$, then $eR\oplus fR$ is a summand of $R$.

maximal right ideals are two-sided

$R$ is Grothendieck and J-2

Noetherian and self-injective

$Nil(R)=J(R)$

No nonzero nilpotent elements

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

$ab=0$ implies $ba=0$

For any $x$ in $R$, the right annihilator of $x$ is a summand of $R$. Also known as: principally projective (p.p.) rings.

normal domain in which every element is primal

$R$ is injective as a right module

All finitely generate right ideals are free and of unique rank

$R$ is right semi-Artinian if every nonzero quotient of $R_R$ contains a minimal submodule. (Equivalently, every nonzero quotient has an essential socle.)

$R$ is right semi-Noetherian if every nonzero submodule of $R_R$ has a maximal submodule

$ab=0$ implies $aRb=\{0\}$ for all $a,b\in R$. Also known as: SI condition, zero-insertive rings.

All finitely generated right ideals are projective

$R/J(R)$ is Artinian

$R$ is semilocal and lift/rad

$R$ is semilocal and $J(R)$ is nilpotent

No nonzero nilpotent right ideals

Jacobson radical zero. Also known as: Jacobson semisimple/J-semisimple

$R$ is lift/rad and $R/J(R)$ is von Neumann regular

semiprimitive and Artinian. Also known as: Wedderburn ring

$R$ is a direct sum of right ideals whose submodules are linearly ordered

Only ideals are the two trivial ideals

$R$ is simple and Artinian

The right socle is a minimal right ideal

If $T$ is a right ideal of $R$, and $f:T\to R$ such that $f(T)$ is simple, then $f$ can be extended to $R\to R$

If $xa+b=1$, there is a $y$ such that $a+yb$ is a unit

Every matrix ring of $R$ is Dedekind finite

For all $x\in R$, $xR\supseteq x^2R\supseteq x^3R\supseteq \ldots$ terminates

The only idempotents are 0 and 1

von Neumann regular and reduced

$abc=0$ implies $acb=0$

Has right T-nilpotent Jacobson radical. Recall a nonempty subset $S$ of $R$ is right T-nilpotent if, for every sequence $\{r_i\mid i\in\mathbb N\}\subseteq S$, there exists an $n$ such that $r_nr_{n-1}\cdots r_0=0$. (The sequence of products is reversed for left T-nilpotent.)

$R/J(R)$ von Neumann regular

$R/J(R)$ simple

$R/J(R)$ simple Artinian

$R$ 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 Generator Property": For all $a,b\in R$, $aR=bR$ implies $a=bu$ for a unit $u\in R$.

All nonzero right ideals essential

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

$R$ is a domain whose right ideals are linearly ordered.

The right ideals of $R$ are linearly ordered. Also known as: right chain ring, right valuation ring

For every $x$, there exists a unit $u$ such that $x=xux$

All finitely generated algebras over $R$ are catenary

every finitely generated integral domain over $R$ is Japanese

"Villamayor ring": The Jacobson radical of every right module is zero

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.)

For every $x$, there exists $y$ such that $x=xyx$

For all $a\in R$, there exists a unit $u$ and idempotent $e$ such that $a-e-u\in (1-e)Ra$. This is left-right symmetric.

$R$ is $I_0$ and $J(R)$ is a nil ideal