Description: Triangular ring $\begin{bmatrix}Z&Q \\ 0&Q\end{bmatrix}$ where $Z$ is a commutative Noetherian domain not equal to its field of fractions $Q$.

Notes:

Keywords triangular ring

Reference(s):

This ring has the following properties:

2-primal
ACC annihilator (right)
ACC principal (right)
coherent (left)
coherent (right)
connected
DCC annihilator (left)
Dedekind finite
finite uniform dimension (right)
finitely generated socle (right)
Goldie (right)
hereditary (right)
IBN
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
Noetherian (right)
nonsingular (left)
nonsingular (right)
orthogonally finite
quasi-duo (left)
quasi-duo (right)
Rickart (left)
Rickart (right)
semihereditary (left)
semihereditary (right)
stably finite
T-nilpotent radical (left)
T-nilpotent radical (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
Abelian
Artinian (left)
Artinian (right)
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
clean
cogenerator ring (left)
cogenerator ring (right)
commutative
continuous (left)
continuous (right)
distributive (left)
distributive (right)
division ring
domain
dual (left)
dual (right)
duo (left)
duo (right)
exchange
FI-injective (right)
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
hereditary (left)
Kasch (left)
Kasch (right)
local
Noetherian (left)
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)
principally injective (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)
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly $\pi$-regular
strongly connected
strongly regular
symmetric
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular
Zorn

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

ACC annihilator (left)
ACC principal (left)
Baer
cohopfian (left)
cohopfian (right)
CS (left)
CS (right)
DCC annihilator (right)
essential socle (left)
essential socle (right)
FI-injective (left)
finite uniform dimension (left)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Goldie (left)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principally injective (left)
quasi-continuous (left)
quasi-continuous (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
weakly clean