Description: The quotient $\mathbb Z[x]/(x^2-1)$ over the integers $\mathbb Z$.

Notes:

Keywords quotient ring polynomial ring

Reference(s):

This ring has the following properties:

2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
coherent (left)
coherent (right)
commutative
connected
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
duo (left)
duo (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
IBN
NI (nilpotents form an ideal)
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
quasi-duo (left)
quasi-duo (right)
reduced
reversible
semicommutative (SI condition, zero-insertive)
semiprime
stably finite
strongly connected
symmetric

The ring lacks the following properties:

$\pi$-regular
$I_0$
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
clean
cogenerator ring (left)
cogenerator ring (right)
continuous (left)
continuous (right)
division ring
domain
essential socle (left)
essential socle (right)
exchange
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
local
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (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
semiperfect
semiprimary
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly $\pi$-regular
strongly regular
top simple
top simple Artinian
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:

Baer
Bezout (left)
Bezout (right)
cohopfian (left)
cohopfian (right)
CS (left)
CS (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
lift/rad
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
principal ideal ring (left)
principal ideal ring (right)
quasi-continuous (left)
quasi-continuous (right)
Rickart (left)
Rickart (right)
semihereditary (left)
semihereditary (right)
semilocal
semiprimitive
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
uniform (left)
uniform (right)
weakly clean