Description: (quite complex) separate page?

Notes:

Reference(s):

- Chatters, A.W. Hajarnavis, C.R., Rings With Chain Conditions, , (1980). Bergman'S Example Pp 27-30

This ring has the following properties:

connected
Dedekind finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
IBN
lift/rad
nil radical
nilpotent radical
orthogonally finite
prime
primitive (left)
primitive (right)
semiprime
semiprimitive
stably finite
T-nilpotent radical (left)
T-nilpotent radical (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
2-primal
Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
clean
commutative
continuous (left)
continuous (right)
division ring
domain
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
finite
finitely cogenerated (left)
finitely cogenerated (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
local
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (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-Frobenius
reduced
reversible
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprimary
semiregular
semisimple
serial (left)
serial (right)
simple Artinian
strongly $\pi$-regular
strongly regular
symmetric
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:

Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Bezout (left)
Bezout (right)
cogenerator ring (left)
cogenerator ring (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
FI-injective (left)
FI-injective (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
fully prime
fully semiprime
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
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
simple
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
strongly connected
top simple
uniform (left)
uniform (right)
V ring (left)
V ring (right)
weakly clean