This is the comprehensive list of ring properties in the database.
Name  Definition  % Complete 

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

$h$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 

$I_0$ 
Every right ideal not contained in $J(R)$ contains a nonzero idempotent. Also known as: semipotent. 

2primal 
All minimal prime ideals are completely prime 

?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 

Abelian 
All idempotents are central 

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

ACC principal 
Ascending chain condition on principal right ideals 

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 

antiautomorphic 
There exists an antiisomorphism of $R$ into itself. 

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.) 

Armendariz 
$R$ 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 $i,j$. 

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

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

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

Bezout 
Finitely generated right ideals are cyclic 

Bezout domain 
Right Bezout domain 

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

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. 

clean 
Every element is the sum of a unit and an idempotent 

cogenerator ring 
The ring cogenerates its category of right modules 

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

coherent 
Every finitely generated right ideal is finitely presented 

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

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

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. 

compressible 
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) 

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

countable 
The underlying set of the ring is countable. 

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

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

Dedekind domain 
A domain whose ideals are projective modules 

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

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

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

distributive 
Lattice of right ideals is a distributive lattice 

division ring 
All nonzero elements are units. Also known as: skewfields, sfields 

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

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

duo 
All right ideals are twosided ideals 

essential socle 
Essential right socle 

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 

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

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

FIinjective 
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 $R\to R$ 

field 
A commutative division ring 

finite 
$R$ has only finitely many elements 

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

finitely cogenerated 
The ring is finitely cogenerated as a right module 

finitely generated socle 
Finitely generated right socle 

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

free ideal ring 
All right ideals are free modules and of unique rank 

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

fully prime 
all proper ideals are prime 

fully semiprime 
all proper ideals are semiprime 

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

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

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 $R[x]$.) 

hereditary 
All right ideals are projective 

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

IC ring 
"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 leftright symmetric.) 

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

involutive 
There exists an antiautomorphism $\tau:R\to R$ such that $\tau^2$ is the identity map on $R$. 

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 

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

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. 

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

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

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

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

max ring 
(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. 

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 $R$. 

McCoy 
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 $fr=0$. 

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 

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

nil radical 
Has nil Jacobson radical 

nilpotent radical 
Has nilpotent Jacobson radical 

Noetherian 
Ascending chain condition on right ideals 

nonsingular 
The right singular ideal is zero 

nonzero socle 
Nonzero right socle 

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 

Ore domain 
Domain satisfying right Ore condition 

Ore ring 
Satisfies right Ore conditions 

orthogonally finite 
families of orthogonal idempotents are all finite 

PCI ring 
(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. 

perfect 
$R$ is semilocal and $J(R)$ is right Tnilpotent 

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

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

polynomial identity 
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 

potent 
The ring is $I_0$ and lift/rad 

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

prime 
The product of nonzero right ideals is nonzero 

primitive 
There exists a faithful simple right $R$ module 

principal ideal domain 
Domain which is a principal right ideal ring 

principal ideal ring 
All right ideals are cyclic 

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

Prufer domain 
A domain whose finitely generated ideals are projective modules 

pseudoFrobenius 
$R$ is right selfinjective and finitely cogenerated as a right module 

Pythagorean field 
The sum of two squares is a square 

quadratically closed field 
Every element is a square of another element 

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

quasiduo 
maximal right ideals are twosided 

quasiexcellent 
$R$ is Grothendieck and J2 

quasiFrobenius 
Noetherian and selfinjective 

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

reduced 
No nonzero nilpotent elements 

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 

reversible 
$ab=0$ implies $ba=0$ 

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

Schreier domain 
normal domain in which every element is primal 

selfinjective 
$R$ is injective as a right module 

semi free ideal ring 
All finitely generate right ideals are free and of unique rank 

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

semiNoetherian 
$R$ is right semiNoetherian if every nonzero submodule of $R_R$ has a maximal submodule 

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

semihereditary 
All finitely generated right ideals are projective 

semilocal 
$R/J(R)$ is Artinian 

semiperfect 
$R$ is semilocal and lift/rad 

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

semiprime 
No nonzero nilpotent right ideals 

semiprimitive 
Jacobson radical zero. Also known as: Jacobson semisimple/Jsemisimple 

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

semisimple 
semiprimitive and Artinian. Also known as: Wedderburn ring 

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

simple 
Only ideals are the two trivial ideals 

simple Artinian 
$R$ is simple and Artinian 

simple socle 
The right socle is a minimal right ideal 

simpleinjective 
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 $R\to R$ 

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

stably finite 
Every matrix ring of $R$ is Dedekind finite 

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

strongly connected 
The only idempotents are 0 and 1 

strongly regular 
von Neumann regular and reduced 

symmetric 
$abc=0$ implies $acb=0$ 

Tnilpotent radical 
Has right Tnilpotent Jacobson radical. Recall a nonempty subset $S$ of $R$ is right Tnilpotent if, for every sequence $\{r_i\mid i\in\mathbb N\}\subseteq S$, there exists an $n$ such that $r_nr_{n1}\cdots r_0=0$. (The sequence of products is reversed for left Tnilpotent.) 

top regular 
$R/J(R)$ von Neumann regular 

top simple 
$R/J(R)$ simple 

top simple Artinian 
$R/J(R)$ simple Artinian 

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.) 

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

uniform 
All nonzero right ideals essential 

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

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

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

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

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

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

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

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.) 

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

weakly clean 
For all $a\in R$, there exists a unit $u$ and idempotent $e$ such that $aeu\in (1e)Ra$. This is leftright symmetric. 

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