Ring $R_{ 214 }$

ACCP ring with non-ACCP polynomial ring

Description:

Let $K$ be a field and $X_1, X_2, \ldots$ be indeterminates over $K$. Let $R_0 = K[X_1, X_2, \ldots] / (\{X_n(X_{n-1} - X_n) \mid n \geq 2\})$. Let $x_n$ denote the image of $X_n$ in $R_0$. Let $R$ be the localization of $R_0$ at the ideal $(x_1, x_2, \ldots)$.

Keywords localization polynomial ring quotient ring

Reference(s):

  • W. J. Heinzer and D. C. Lantz. ACCP in polynomial rings: a counterexample. (1994) @ main example


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