Description: The ring of $n \times n$ matrices over a finite field $F$ whose entries below the diagonal are all zero.

Notes:

Keywords matrix ring

Reference(s):

This ring has the following properties:

$\pi$-regular
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)
I_0
IBN
lift/rad
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
nonzero socle (right)
orthogonally finite
perfect (left)
perfect (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
Zorn

The ring lacks the following properties:

Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
commutative
connected
distributive (left)
distributive (right)
division ring
domain
dual (left)
dual (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
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)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
self-injective (left)
self-injective (right)
semi free ideal ring
semiprime
semiprimitive
semisimple
simple
simple Artinian
simple socle (left)
simple socle (right)
strongly regular
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:

2-primal
Abelian
Baer
Bezout (left)
Bezout (right)
continuous (left)
continuous (right)
duo (left)
duo (right)
FI-injective (left)
FI-injective (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
quasi-duo (left)
quasi-duo (right)
reversible
simple-injective (left)
simple-injective (right)
symmetric
top simple