Description: Let $S$ be the ring of linear transformations of a countable dimensional vector space over a field. This is known to have exactly one nontrivial ideal $M$. The ring is $S/M$

Notes:

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
Bezout (left)
Bezout (right)
clean
coherent (left)
coherent (right)
connected
exchange
fully prime
fully semiprime
lift/rad
nil radical
nilpotent radical
nonsingular (left)
nonsingular (right)
prime
primitive (left)
primitive (right)
principally injective (left)
principally injective (right)
Rickart (left)
Rickart (right)
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
simple
T-nilpotent radical (left)
T-nilpotent radical (right)
von Neumann regular
weakly clean
Zorn

The ring lacks the following properties:

2-primal
Abelian
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
commutative
distributive (left)
distributive (right)
division ring
domain
duo (left)
duo (right)
finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
Goldie (left)
Goldie (right)
local
Noetherian (left)
Noetherian (right)
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-duo (left)
quasi-duo (right)
quasi-Frobenius
reduced
reversible
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semisimple
serial (left)
serial (right)
simple Artinian
strongly connected
strongly regular
symmetric
top simple Artinian
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)

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

ACC annihilator (left)
ACC annihilator (right)
Baer
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
dual (left)
dual (right)
essential socle (left)
essential socle (right)
FI-injective (left)
FI-injective (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
hereditary (left)
hereditary (right)
IBN
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)
orthogonally finite
polynomial identity
quasi-continuous (left)
quasi-continuous (right)
self-injective (left)
self-injective (right)
semiregular
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly $\pi$-regular
top regular
top simple
unit regular
V ring (left)
V ring (right)