# Ring properties

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 $(a^n)x(a^n)=a^n$
89%
$h$-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%
$I_0$
Every right ideal not contained in $J(R)$ contains a nonzero idempotent. Also known as: semipotent.
83%
2-primal
All minimal prime ideals are completely prime
87%
?-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%
Abelian
All idempotents are central
89%
ACC annihilator
Ascending chain condition on right annihilators=Descending chain condition on left annihilators
84%
ACC principal
Ascending chain condition on principal right ideals
75%
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%
anti-automorphic
There exists an anti-isomorphism of $R$ into itself.
72%
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%
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 $i,j$.
77%
Artinian
Right Artinian (resp. left Artinian) Descending chain condition on right (resp. left) ideals.
98%
atomic domain
A domain in which nonzero nonunits can be written as a finite product of irreducible elements.
96%
Baer
For any subset $X$ of $R$, the right annihilator of $X$ is a summand of $R$
75%
Bezout
Finitely generated right ideals are cyclic
65%
Bezout domain
Right Bezout domain
91%
Boolean
For every $x\in R$, $x^2=x$.
100%
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%
clean
Every element is the sum of a unit and an idempotent
81%
cogenerator ring
The ring cogenerates its category of right modules
76%
Cohen-Macaulay
Noetherian ring whose localizations at primes all have depth equal to their Krull dimension
86%
coherent
Every finitely generated right ideal is finitely presented
67%
cohopfian
For every $x$ with right annihilator zero, $x$ is a unit
77%
commutative
$xy=yx$ for all $x$ and $y$ in the ring
99%
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%
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)
74%
continuous
$R$ is right CS, and any right ideal isomorphic to a direct summand of $R$ is itself a summand.
71%
countable
The underlying set of the ring is countable.
76%
CS
"Complements are Summands": Every nonzero right ideal is essential in a direct summand of $R_R$.
57.0%
DCC annihilator
Descending chain condition on right annihilators=Ascending chain condition on left annihilators
84%
Dedekind domain
A domain whose ideals are projective modules
98%
Dedekind finite
For all $x,y\in R$, $xy=1$ implies $yx=1$. Also known as: directly finite, von Neumann finite
93%
directly irreducible
The only central idempotents are $0$ and $1$. Also known as: connected (especially for commutative rings)
83%
discrete valuation ring
$R$ is a valuation ring with value group isomorphic to $(\mathbb Z,+)$
99%
distributive
Lattice of right ideals is a distributive lattice
69%
division ring
All nonzero elements are units. Also known as: skew-fields, sfields
99%
domain
No nonzero zero divisors. Also known as: entire rings (mainly for commutative domains)
94%
dual
For every right ideal $B$, $r.ann(l.ann(B))=B$
77%
duo
All right ideals are two-sided ideals
76%
essential socle
Essential right socle
73%
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%
exchange
If $a+b=1$, there exists $r$ and $s$ such that $ar$ and $bs$ are idempotents and $ar+bs=1$
84%
FGC
"Finitely Generated are Cyclic": A commutative ring for which every finitely generated module is a direct sum of cyclic modules.
79%
FI-injective
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 $R\to R$
70%
field
A commutative division ring
100%
finite
$R$ has only finitely many elements
98%
finite uniform dimension
$R$ has finite uniform dimension as a right $R$ module
85%
finitely cogenerated
The ring is finitely cogenerated as a right module
87%
finitely generated socle
Finitely generated right socle
70%
finitely pseudo-Frobenius
Every finitely generated faithful right module is a generator for the category of $R$ modules
39%
free ideal ring
All right ideals are free modules and of unique rank
87%
Frobenius
$R$ is quasi-Frobenius and the right socle is isomorphic to $R/J(R)$
98%
fully prime
all proper ideals are prime
80%
fully semiprime
all proper ideals are semiprime
79%
GCD domain
A domain having a gcd for every pair of elements
95%
Goldie
$R$ has finite right uniform dimension and ACC on right annihilators
86%
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%
hereditary
All right ideals are projective
68%
IBN
(Invariant Basis Number) If $R^n$ is isomorphic to $R^m$, then $n=m$
94%
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 left-right symmetric.)
87%
Ikeda-Nakayama
The left annihilator of the intersection of two right ideals, is the sum of their left annihilators
56.0%
involutive
There exists an anti-automorphism $\tau:R\to R$ such that $\tau^2$ is the identity map on $R$.
72%
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%
Kasch
Every simple right module is isomorphic to a minimal right ideal of $R$
70%
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%
Idempotents of $R/J(R)$ lift to idempotents in R
81%
linearly compact
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.
81%
local
$R/J(R)$ is a division ring
89%
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%
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.
56.0%
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%
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 $fr=0$.
68%
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%
NI ring
"Nilpotents form an Ideal": The set of nilpotent elements in $R$ forms an ideal.
85%
79%
81%
Noetherian
Ascending chain condition on right ideals
91%
nonsingular
The right singular ideal is zero
80%
nonzero socle
Nonzero right socle
74%
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%
Ore domain
Domain satisfying right Ore condition
93%
Ore ring
Satisfies right Ore conditions
80%
orthogonally finite
families of orthogonal idempotents are all finite
89%
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.
92%
perfect
$R$ is semilocal and $J(R)$ is right T-nilpotent
96%
perfect field
A field over which every irreducible polynomial has distinct roots.
100%
periodic
For every element $x\in R$, there exists a natural number $n_x> 1$ such that $x^{n_x}=x$.
100%
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
70%
potent
The ring is $I_0$ and lift/rad
83%
primary
$R$ semiprimary and $R/J(R)$ simple
96%
prime
The product of nonzero right ideals is nonzero
89%
primitive
There exists a faithful simple right $R$ module
82%
principal ideal domain
Domain which is a principal right ideal ring
93%
principal ideal ring
All right ideals are cyclic
83%
principally injective
homomorphisms from principal right ideals of the ring into the ring extend to endomorphisms of the ring
73%
Prufer domain
A domain whose finitely generated ideals are projective modules
95%
pseudo-Frobenius
$R$ is right self-injective and finitely cogenerated as a right module
97%
Pythagorean field
The sum of two squares is a square
99%
Every element is a square of another element
100%
quasi-continuous
$R$ is right CS, and if $e,f$ are idempotents with $eR\cap fR=\{0\}$, then $eR\oplus fR$ is a summand of $R$.
58.0%
quasi-duo
maximal right ideals are two-sided
67%
quasi-excellent
$R$ is Grothendieck and J-2
81%
quasi-Frobenius
Noetherian and self-injective
98%
$Nil(R)=J(R)$
81%
reduced
No nonzero nilpotent elements
94%
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%
reversible
$ab=0$ implies $ba=0$
85%
Rickart
For any $x$ in $R$, the right annihilator of $x$ is a summand of $R$. Also known as: principally projective (p.p.) rings.
86%
Schreier domain
normal domain in which every element is primal
95%
self-injective
$R$ is injective as a right module
88%
semi free ideal ring
All finitely generate right ideals are free and of unique rank
88%
semi-Artinian
$R$ is right semi-Artinian if every nonzero quotient of $R_R$ contains a minimal submodule. (Equivalently, every nonzero quotient has an essential socle.)
87%
semi-Noetherian
$R$ is right semi-Noetherian if every nonzero submodule of $R_R$ has a maximal submodule
52%
semicommutative
$ab=0$ implies $aRb=\{0\}$ for all $a,b\in R$. Also known as: SI condition, zero-insertive rings.
88%
semihereditary
All finitely generated right ideals are projective
75%
semilocal
$R/J(R)$ is Artinian
85%
semiperfect
$R$ is semilocal and lift/rad
90%
semiprimary
$R$ is semilocal and $J(R)$ is nilpotent
97%
semiprime
No nonzero nilpotent right ideals
93%
semiprimitive
Jacobson radical zero. Also known as: Jacobson semisimple/J-semisimple
84%
semiregular
$R$ is lift/rad and $R/J(R)$ is von Neumann regular
86%
semisimple
semiprimitive and Artinian. Also known as: Wedderburn ring
99%
serial
$R$ is a direct sum of right ideals whose submodules are linearly ordered
83%
simple
Only ideals are the two trivial ideals
84%
simple Artinian
$R$ is simple and Artinian
99%
simple socle
The right socle is a minimal right ideal
68%
simple-injective
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 $R\to R$
54%
stable range 1
If $xa+b=1$, there is a $y$ such that $a+yb$ is a unit
64%
stably finite
Every matrix ring of $R$ is Dedekind finite
88%
strongly $\pi$-regular
For all $x\in R$, $xR\supseteq x^2R\supseteq x^3R\supseteq \ldots$ terminates
88%
strongly connected
The only idempotents are 0 and 1
88%
strongly regular
von Neumann regular and reduced
99%
symmetric
$abc=0$ implies $acb=0$
84%
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.)
81%
top regular
$R/J(R)$ von Neumann regular
80%
top simple
$R/J(R)$ simple
75%
top simple Artinian
$R/J(R)$ simple Artinian
85%
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%
UGP ring
"Unique Generator Property": For all $a,b\in R$, $aR=bR$ implies $a=bu$ for a unit $u\in R$.
59%
uniform
All nonzero right ideals essential
80%
unique factorization domain
A domain in which every nonzero nonunit is a product of irreducible elements, unique up to equivalence
94%
uniserial domain
$R$ is a domain whose right ideals are linearly ordered.
96%
uniserial ring
The right ideals of $R$ are linearly ordered. Also known as: right chain ring, right valuation ring
89%
unit regular
For every $x$, there exists a unit $u$ such that $x=xux$
96%
universally catenary
All finitely generated algebras over $R$ are catenary
47%
universally Japanese
every finitely generated integral domain over $R$ is Japanese
50%
V ring
"Villamayor ring": The Jacobson radical of every right module is zero
74%
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%
von Neumann regular
For every $x$, there exists $y$ such that $x=xyx$
96%
weakly clean
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.
76%
Zorn
$R$ is $I_0$ and $J(R)$ is a nil ideal
81%