*This ring suggested by: jose.simental*

Description: $K\langle x,y\rangle /(xy - yx - 1)$ where $K$ is a field.

Notes:

Keywords quotient ring free algebra

Reference(s):

This ring has the following properties:

2-primal
Abelian
Baer
coherent (left)
coherent (right)
connected
Dedekind finite
domain
fully prime
fully semiprime
hereditary (left)
hereditary (right)
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
nonsingular (left)
nonsingular (right)
orthogonally finite
prime
primitive (left)
primitive (right)
reduced
reversible
Rickart (left)
Rickart (right)
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
simple
strongly connected
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
Artinian (left)
Artinian (right)
clean
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
commutative
continuous (left)
continuous (right)
division ring
dual (left)
dual (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite
finitely cogenerated (left)
finitely cogenerated (right)
Frobenius
Kasch (left)
Kasch (right)
local
nonzero socle (left)
nonzero socle (right)
perfect (left)
perfect (right)
primary
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
semilocal
semiperfect
semiprimary
semiregular
semisimple
serial (left)
serial (right)
simple Artinian
simple socle (left)
simple socle (right)
strongly $\pi$-regular
strongly regular
top regular
top simple Artinian
unit regular
valuation ring (left)
valuation ring (right)
von Neumann regular
Zorn

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

ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
distributive (left)
distributive (right)
duo (left)
duo (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
Goldie (left)
Goldie (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Noetherian (left)
Noetherian (right)
Ore domain (left)
Ore domain (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
semi free ideal ring
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
top simple
uniform (left)
uniform (right)
V ring (left)
V ring (right)
weakly clean