Description: polynomial ring over the real numbers mod the ideal generated by $x^2, y^2, xz,yz,z^2-xy$

Notes: Krull dimension 0

Keywords quotient ring polynomial ring

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
clean
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
commutative
connected
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
IBN
Kasch (left)
Kasch (right)
lift/rad
local
NI (nilpotents form an ideal)
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
perfect (left)
perfect (right)
polynomial identity
primary
quasi-duo (left)
quasi-duo (right)
reversible
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semiregular
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
weakly clean
Zorn

The ring lacks the following properties:

Baer
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
division ring
domain
free ideal ring (left)
free ideal ring (right)
fully prime
fully semiprime
hereditary (left)
hereditary (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
reduced
Rickart (left)
Rickart (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly regular
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular

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

cogenerator ring (left)
cogenerator ring (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
FI-injective (left)
FI-injective (right)
finite
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Frobenius
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
uniform (left)
uniform (right)