Description: Let $k$ be a field. Let $S$ be the power series ring over $k$ in two noncommuting variables $x,y$. The ring is $R=S/(y^2,yx)$

Notes:

Keywords quotient ring power series ring

Reference(s):

This ring has the following properties:

$I_0$
Abelian
clean
connected
Dedekind finite
exchange
IBN
Kasch (right)
lift/rad
local
nonzero socle (right)
orthogonally finite
quasi-duo (left)
quasi-duo (right)
semilocal
semiperfect
semiregular
stable range 1
stably finite
strongly connected
top regular
top simple
top simple Artinian
weakly clean

The ring lacks the following properties:

Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
commutative
division ring
domain
dual (left)
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
Kasch (left)
nilpotent radical
Ore domain (left)
Ore domain (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (right)
semi free ideal ring
semiprimary
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:

$\pi$-regular
2-primal
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Baer
Bezout (left)
Bezout (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
distributive (left)
distributive (right)
dual (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
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)
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
NI (nilpotents form an ideal)
nil radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
Ore ring (left)
Ore ring (right)
perfect (left)
perfect (right)
polynomial identity
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
reduced
reversible
Rickart (left)
Rickart (right)
self-injective (left)
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semiprime
serial (left)
serial (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
strongly $\pi$-regular
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)
Zorn