Ring $R_{ 68 }$

McGovern's commutative Zorn ring that isn't clean

Description:

Let $X$ be a zero-dimensional topological space such that the Stone-Čech compactification $\beta X$ is not zero-dimensional. The required ring is $R=C(X)$, the set of continuous functions from $X$ into $\mathbb R$. (A topological space is called zero-dimensional if it has a base of clopen sets.)

Keywords ring of functions

Reference(s):

  • W. W. McGovern. Clean semiprime f-rings with bounded inversion. (2003) @ Example 20


Known Properties
Name
ACC annihilator
ACC principal
almost maximal ring
arithmetical
Baer
Bezout
catenary
cogenerator ring
coherent
cohopfian
countable
CS
DCC annihilator
distributive
essential socle
FGC
FI-injective
finitely generated socle
finitely pseudo-Frobenius
hereditary
Ikeda-Nakayama
J-0
J-1
J-2
Jacobson
Kasch
max ring
nonzero socle
normal
orthogonally finite
principally injective
quasi-continuous
Rickart
semi-Noetherian
semihereditary
simple socle
simple-injective
stable range 1
UGP ring
universally catenary
universally Japanese
weakly clean
$\pi$-regular
$h$-local domain
?-ring
algebraically closed field
almost Dedekind domain
almost maximal domain
almost maximal valuation ring
analytically normal
analytically unramified
Archimedean field
Artinian
atomic domain
Bezout domain
Boolean
characteristic 0 field
clean
Cohen-Macaulay
complete discrete valuation ring
complete local
continuous
Dedekind domain
directly irreducible
discrete valuation ring
division ring
domain
dual
Euclidean domain
Euclidean field
excellent
exchange
field
finite
finite uniform dimension
finitely cogenerated
free ideal ring
Frobenius
fully prime
fully semiprime
GCD domain
Goldie
Goldman domain
Gorenstein
Grothendieck
Henselian local
Krull domain
linearly compact
local
local complete intersection
maximal ring
maximal valuation ring
Mori domain
N-1
N-2
Nagata
Noetherian
normal domain
ordered field
Ore domain
PCI ring
perfect
perfect field
periodic
primary
prime
primitive
principal ideal domain
principal ideal ring
Prufer domain
pseudo-Frobenius
Pythagorean field
quadratically closed field
quasi-excellent
quasi-Frobenius
regular
regular local
Schreier domain
self-injective
semi free ideal ring
semi-Artinian
semilocal
semiperfect
semiprimary
semiregular
semisimple
serial
simple
simple Artinian
strongly $\pi$-regular
strongly connected
strongly regular
top regular
top simple
top simple Artinian
torch
uniform
unique factorization domain
uniserial domain
uniserial ring
unit regular
V ring
valuation domain
valuation ring
von Neumann regular
$I_0$
2-primal
Abelian
anti-automorphic
Armendariz
commutative
compressible
Dedekind finite
duo
IBN
IC ring
involutive
lift/rad
McCoy
NI ring
nil radical
nilpotent radical
nonsingular
Ore ring
polynomial identity
potent
quasi-duo
rad-nil
reduced
reversible
semicommutative
semiprime
semiprimitive
stably finite
symmetric
T-nilpotent radical
Zorn
Legend
  • = has the property
  • = does not have the property
  • = information not in database
Name Measure
composition length left: $\infty$right: $\infty$
Name Description
Jacobson radical $\{0\}$
Left singular ideal $\{0\}$
Nilpotents $\{0\}$
Right singular ideal $\{0\}$