Description: The ring of $n \times n$ matrices with entries from a field $F$, $n$ a natural number greater than $1$, and $F$ an infinite field.

Notes:

Keywords matrix ring

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Baer
Bezout (left)
Bezout (right)
clean
cogenerator ring (left)
cogenerator ring (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
dual (left)
dual (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
lift/rad
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
perfect (left)
perfect (right)
polynomial identity
primary
prime
primitive (left)
primitive (right)
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-Frobenius
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly $\pi$-regular
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
unit regular
V ring (left)
V ring (right)
von Neumann regular
weakly clean
Zorn

The ring lacks the following properties:

2-primal
Abelian
Bezout domain (left)
Bezout domain (right)
commutative
distributive (left)
distributive (right)
division ring
domain
duo (left)
duo (right)
finite
free ideal ring (left)
free ideal ring (right)
local
NI (nilpotents form an ideal)
Ore domain (left)
Ore domain (right)
principal ideal domain (left)
principal ideal domain (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
semi free ideal ring
semicommutative (SI condition, zero-insertive)
simple socle (left)
simple socle (right)
strongly connected
strongly regular
symmetric
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)

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