Ring $R_{ 100 }$

Nagata's Noetherian infinite Krull dimension ring

Description:

Let $\{I_i\mid i\in \mathbb N\}$ be a partition of $\mathbb N$ such that $|I_i|<|I_{i+1}|$. From $\mathbb Q[x_0, x_1,\ldots]$, consider the prime ideals $P_i=(\{x_j\mid j\in I_i\})$. The ring $R$ is the semilocalization at the intersection of the complements of the $P_i$'s.

Keywords localization polynomial ring

Reference(s):

  • M. Nagata. Local rings. (1962) @ p 203
  • T.-Y. Lam. Lectures on modules and rings. (2012) @ Example 5.95 pp 200-201


Known Properties
Name
$h$-local domain
almost maximal domain
almost maximal ring
countable
excellent
finitely pseudo-Frobenius
Goldman domain
Grothendieck
J-0
J-1
J-2
Jacobson
lift/rad
max ring
N-2
Nagata
nil radical
nilpotent radical
quasi-excellent
rad-nil
semiprimitive
stable range 1
T-nilpotent radical
top regular
UGP ring
universally Japanese
$\pi$-regular
$I_0$
?-ring
algebraically closed field
almost Dedekind domain
almost maximal valuation ring
analytically normal
analytically unramified
Archimedean field
arithmetical
Artinian
Bezout
Bezout domain
Boolean
characteristic 0 field
clean
cogenerator ring
cohopfian
complete discrete valuation ring
complete local
continuous
Dedekind domain
discrete valuation ring
distributive
division ring
dual
essential socle
Euclidean domain
Euclidean field
exchange
FGC
FI-injective
field
finite
finitely cogenerated
free ideal ring
Frobenius
fully prime
fully semiprime
Henselian local
hereditary
Kasch
linearly compact
local
local complete intersection
maximal ring
maximal valuation ring
nonzero socle
ordered field
PCI ring
perfect
perfect field
periodic
potent
primary
primitive
principal ideal domain
principal ideal ring
principally injective
Prufer domain
pseudo-Frobenius
Pythagorean field
quadratically closed field
quasi-Frobenius
regular local
self-injective
semi free ideal ring
semi-Artinian
semihereditary
semilocal
semiperfect
semiprimary
semiregular
semisimple
serial
simple
simple Artinian
simple socle
strongly $\pi$-regular
strongly regular
top simple
top simple Artinian
torch
uniserial domain
uniserial ring
unit regular
V ring
valuation domain
valuation ring
von Neumann regular
weakly clean
Zorn
2-primal
Abelian
ACC annihilator
ACC principal
anti-automorphic
Armendariz
atomic domain
Baer
catenary
Cohen-Macaulay
coherent
commutative
compressible
CS
DCC annihilator
Dedekind finite
directly irreducible
domain
duo
finite uniform dimension
finitely generated socle
GCD domain
Goldie
Gorenstein
IBN
IC ring
Ikeda-Nakayama
involutive
Krull domain
McCoy
Mori domain
N-1
NI ring
Noetherian
nonsingular
normal
normal domain
Ore domain
Ore ring
orthogonally finite
polynomial identity
prime
quasi-continuous
quasi-duo
reduced
regular
reversible
Rickart
Schreier domain
semi-Noetherian
semicommutative
semiprime
simple-injective
stably finite
strongly connected
symmetric
uniform
unique factorization domain
universally catenary
Legend
  • = has the property
  • = does not have the property
  • = information not in database
Name Measure
cardinality $\aleph_0$
global dimension left: $\infty$right: $\infty$
Krull dimension (classical) $\infty$
Name Description
Idempotents $\{0,1\}$
Left singular ideal $\{0\}$
Left socle $\{0\}$
Nilpotents $\{0\}$
Right singular ideal $\{0\}$
Right socle $\{0\}$
Zero divisors $\{0\}$