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:

ACC annihilator (right)
ACC principal (right)
coherent (left)
coherent (right)
DCC annihilator (left)
Dedekind finite
finite uniform dimension (right)
finitely generated socle (right)
Goldie (right)
hereditary (right)
IBN
Noetherian (right)
nonsingular (left)
nonsingular (right)
orthogonally finite
Rickart (left)
Rickart (right)
semihereditary (left)
semihereditary (right)
stably finite

The ring lacks the following properties:

$\pi$-regular
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
commutative
distributive (left)
distributive (right)
division ring
domain
dual (left)
dual (right)
FI-injective (right)
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
hereditary (left)
Kasch (left)
Kasch (right)
Noetherian (left)
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
self-injective (left)
self-injective (right)
semi free ideal ring
semiprimary
semiprimitive
semisimple
simple
simple Artinian
strongly $\pi$-regular
strongly regular
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
ACC annihilator (left)
ACC principal (left)
Baer
Bezout (left)
Bezout (right)
clean
cohopfian (left)
cohopfian (right)
connected
continuous (left)
continuous (right)
DCC annihilator (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
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)
fully prime
fully semiprime
Goldie (left)
I_0
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
local
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
polynomial identity
prime
principal ideal ring (right)
principally injective (left)
quasi-duo (left)
quasi-duo (right)
reversible
semilocal
semiperfect
semiprime
semiregular
serial (left)
serial (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
Zorn