Description: $2 \times 2$ ring of upper triangular matrices over the field of two elements.

Notes: Lattice of right ideals isomorphic to the $M_3$ diamond lattice

Keywords matrix ring

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
2-primal
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
clean
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
essential socle (left)
essential socle (right)
exchange
finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
IBN
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
nonzero socle (right)
orthogonally finite
perfect (left)
perfect (right)
polynomial identity
quasi-duo (left)
quasi-duo (right)
Rickart (left)
Rickart (right)
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprimary
semiregular
serial (left)
serial (right)
stable range 1
stably finite
strongly $\pi$-regular
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
weakly clean
Zorn

The ring lacks the following properties:

Abelian
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
commutative
distributive (left)
distributive (right)
division ring
domain
dual (left)
dual (right)
duo (left)
duo (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
local
Ore domain (left)
Ore domain (right)
primary
prime
primitive (left)
primitive (right)
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
self-injective (left)
self-injective (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semiprime
semiprimitive
semisimple
simple
simple Artinian
simple socle (left)
simple socle (right)
strongly connected
strongly regular
symmetric
top simple Artinian
uniform (left)
uniform (right)
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular

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

Baer
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
FI-injective (left)
FI-injective (right)
Ore ring (left)
Ore ring (right)
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
simple-injective (left)
simple-injective (right)
top simple