Description: Triangular ring $\begin{bmatrix}\mathbb Z&\mathbb Z/(2)\\ 0&\mathbb Z/(2)\end{bmatrix}$

Notes:

Keywords triangular ring

Reference(s):

This ring has the following properties:

ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
coherent (left)
coherent (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
IBN
Noetherian (left)
Noetherian (right)
nonsingular (right)
orthogonally finite
stably finite

The ring lacks the following properties:

Baer
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
commutative
distributive (right)
division ring
domain
dual (left)
dual (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
hereditary (left)
hereditary (right)
Kasch (left)
Kasch (right)
nonsingular (left)
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
reduced
Rickart (left)
self-injective (left)
self-injective (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly regular
unit regular
V ring (left)
V ring (right)
valuation ring (right)
von Neumann regular

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

$\pi$-regular
2-primal
Abelian
Artinian (left)
Artinian (right)
Bezout (left)
Bezout (right)
clean
cohopfian (left)
cohopfian (right)
connected
continuous (left)
continuous (right)
distributive (left)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite
finitely cogenerated (left)
finitely cogenerated (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
I_0
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
local
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
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-duo (left)
quasi-duo (right)
reversible
Rickart (right)
semilocal
semiperfect
semiprimary
semiregular
serial (left)
serial (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
strongly $\pi$-regular
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
valuation ring (left)
Zorn