Description: The quotient of the real polynomial ring $\mathbb R[x]$ by the ideal generated by $x^2$

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)
Bezout (left)
Bezout (right)
clean
cogenerator ring (left)
cogenerator ring (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
commutative
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (left)
distributive (right)
dual (left)
dual (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Frobenius
Goldie (left)
Goldie (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
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
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
quasi-Frobenius
reversible
self-injective (left)
self-injective (right)
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semiregular
serial (left)
serial (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
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
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)
weakly clean
Zorn

The ring lacks the following properties:

Baer
Bezout domain (left)
Bezout domain (right)
division ring
domain
finite
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)
reduced
Rickart (left)
Rickart (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly regular
unit regular
V ring (left)
V ring (right)
von Neumann regular

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