Description: Quotient of the free algebra $\mathbb Z\langle x,y\rangle$ by the ideal $(y^2,yx)$.

Notes:

Keywords quotient ring free algebra

Reference(s):

- Lam, Tsit-Yuen., A First Course In Noncommutative Rings, Vol. 131. Springer, (2001). P 21

This ring has the following properties:

ACC annihilator (left)
ACC principal (left)
coherent (left)
DCC annihilator (right)
Dedekind finite
finite uniform dimension (left)
finitely generated socle (left)
Goldie (left)
IBN
lift/rad
nil radical
nilpotent radical
Noetherian (left)
nonsingular (left)
orthogonally finite
stably finite
T-nilpotent radical (left)
T-nilpotent radical (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
clean
commutative
continuous (left)
continuous (right)
distributive (left)
division ring
domain
exchange
FI-injective (left)
finite
finite uniform dimension (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (right)
hereditary (left)
hereditary (right)
local
Noetherian (right)
nonsingular (right)
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (right)
principally injective (left)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
Rickart (right)
self-injective (left)
self-injective (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly $\pi$-regular
strongly regular
top regular
top simple Artinian
uniform (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:

2-primal
Abelian
ACC annihilator (right)
ACC principal (right)
Bezout (left)
Bezout (right)
cogenerator ring (left)
cogenerator ring (right)
coherent (right)
cohopfian (left)
cohopfian (right)
connected
CS (left)
CS (right)
DCC annihilator (left)
distributive (right)
dual (left)
dual (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
FI-injective (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
NI (nilpotents form an ideal)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal ring (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reversible
Rickart (left)
semicommutative (SI condition, zero-insertive)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
strongly connected
symmetric
top simple
uniform (left)
weakly clean