*This ring suggested by: Watson*

Description: $\Bbb Q[X,Y]$ localized at the prime ideal $(X,Y)$.

Notes: A local ring with an infinite prime spectrum. It has Krull dimension $2$.

Keywords localization polynomial ring

Reference(s):

This ring has the following properties:

$I_0$
2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Baer
clean
coherent (left)
coherent (right)
commutative
connected
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
domain
duo (left)
duo (right)
exchange
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
local
NI (nilpotents form an ideal)
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
prime
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprime
semiregular
stable range 1
stably finite
strongly connected
symmetric
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
weakly clean

The ring lacks the following properties:

$\pi$-regular
Artinian (left)
Artinian (right)
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
division ring
dual (left)
dual (right)
essential socle (left)
essential socle (right)
FI-injective (left)
FI-injective (right)
finite
finitely cogenerated (left)
finitely cogenerated (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Kasch (left)
Kasch (right)
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
perfect (left)
perfect (right)
primary
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
semi free ideal ring
semiprimary
semiprimitive
semisimple
serial (left)
serial (right)
simple
simple Artinian
simple socle (left)
simple socle (right)
strongly $\pi$-regular
strongly regular
T-nilpotent radical (left)
T-nilpotent radical (right)
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular
Zorn

We don't know if the ring has or lacks the following properties: