Description: Let $S$ be von Neumann regular which has an ideal $I$ which is not a direct summand of $S_S$. Let $R$ be $S/I$. Form the triangular ring $T=\begin{bmatrix}R&R\\ 0& S\end{bmatrix}$. $T$ is the ring.

Notes:

Keywords quotient ring triangular ring

Reference(s):

This ring has the following properties:

The ring lacks the following properties:

Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
commutative
connected
distributive (left)
division ring
domain
finite
finite uniform dimension (left)
finite uniform dimension (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (right)
local
Noetherian (left)
Noetherian (right)
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (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
Rickart (right)
self-injective (left)
semi free ideal ring
semihereditary (right)
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semisimple
serial (left)
serial (right)
simple
simple Artinian
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:

$\pi$-regular
2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Bezout (left)
Bezout (right)
clean
cogenerator ring (left)
cogenerator ring (right)
coherent (right)
cohopfian (left)
cohopfian (right)
continuous (left)
continuous (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (right)
dual (left)
dual (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
FI-injective (left)
FI-injective (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
hereditary (left)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
nonsingular (right)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
principally injective (left)
principally injective (right)
quasi-duo (left)
quasi-duo (right)
reversible
self-injective (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly $\pi$-regular
symmetric
top simple