Description: The ring is $\begin{bmatrix} F_2&F_2\\0&Z_{(2)}\end{bmatrix}$ where the field of two elements $F_2$ is given the natural right $\mathbb Z_{(2)}$ module structure by localizing $\mathbb Z$ at the prime ideal $(2)$

Notes:

Keywords localization triangular ring

Reference(s):

This ring has the following properties:

$I_0$
2-primal
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
clean
coherent (left)
coherent (right)
cohopfian (left)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
exchange
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
IBN
lift/rad
NI (nilpotents form an ideal)
Noetherian (left)
Noetherian (right)
nonzero socle (right)
orthogonally finite
semilocal
semiperfect
semiregular
simple socle (right)
stable range 1
stably finite
top regular
weakly clean

The ring lacks the following properties:

$\pi$-regular
Abelian
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
cohopfian (right)
commutative
distributive (left)
distributive (right)
division ring
domain
duo (left)
duo (right)
essential socle (right)
FI-injective (left)
FI-injective (right)
finite
finitely cogenerated (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
local
nil radical
nilpotent radical
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
reversible
self-injective (left)
self-injective (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semiprimary
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly $\pi$-regular
strongly connected
strongly regular
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top simple
top simple Artinian
uniform (left)
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:

Baer
Bezout (left)
Bezout (right)
cogenerator ring (left)
cogenerator ring (right)
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
dual (left)
dual (right)
essential socle (left)
finitely cogenerated (left)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal ring (left)
principal ideal ring (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
Rickart (left)
Rickart (right)
semihereditary (left)
semihereditary (right)
serial (left)
serial (right)
simple socle (left)
simple-injective (left)
simple-injective (right)