Description: Let $k$ be a division ring, and $R$ be the ring of matrices of the form $\begin{bmatrix} a&0&b&c\\ 0&a&0&d\\ 0&0&a&0\\ 0&0&0&e\end{bmatrix}$

Notes:

Keywords subring 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)
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 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)
IBN
Kasch (right)
lift/rad
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonzero socle (left)
nonzero socle (right)
orthogonally finite
perfect (left)
perfect (right)
semilocal
semiperfect
semiprimary
semiregular
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:

Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
commutative
division ring
domain
dual (left)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Kasch (left)
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 connected
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)
cogenerator ring (right)
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
distributive (left)
distributive (right)
dual (right)
duo (left)
duo (right)
FI-injective (left)
FI-injective (right)
finite
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
NI (nilpotents form an ideal)
nonsingular (left)
nonsingular (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reversible
Rickart (left)
Rickart (right)
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
serial (left)
serial (right)
simple-injective (left)
simple-injective (right)
symmetric
top simple