Ring $R_{ 129 }$

Noetherian ring that is not Grothendieck and not Nagata

Description:

Let $k$ be a field of characteristic $p>0$ such that $[k:k^p]=\infty$. Let $A=k[[x]]$. The required ring is the subring $R$ of elements of $A$ of elements of the form $\sum_0^\infty k_ix^i$ satisfying $[k^p(k_0, k_1,k_2,\ldots): k^p] <\infty$.

Keywords power series ring subring

Reference(s):

  • H. Matsumura. Commutative algebra. (1970) @ Chapter 13 item 34b p 260 (2nd ed)


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