*This ring suggested by: Watson*

Description: Let $F$ be a field, $V$ be an infinite dimensional $F$-vector space, and consider the trivial extension $F(+)V$.

Notes: Radical has square zero. Krull dimension $0$.

Keywords trivial extension

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
2-primal
Abelian
ACC principal (left)
ACC principal (right)
clean
cohopfian (left)
cohopfian (right)
commutative
connected
Dedekind finite
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
IBN
Kasch (left)
Kasch (right)
lift/rad
local
NI (nilpotents form an ideal)
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
perfect (left)
perfect (right)
polynomial identity
primary
quasi-duo (left)
quasi-duo (right)
reversible
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semiregular
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
weakly clean
Zorn

The ring lacks the following properties:

Artinian (left)
Artinian (right)
Baer
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
division ring
domain
finite
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)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
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 (left)
Rickart (right)
self-injective (left)
self-injective (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
semisimple
serial (left)
serial (right)
simple
simple Artinian
simple socle (left)
simple socle (right)
strongly regular
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:

ACC annihilator (left)
ACC annihilator (right)
coherent (left)
coherent (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
FI-injective (left)
FI-injective (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
simple-injective (left)
simple-injective (right)