Description: For a field $F$, the quotient polynomial ring $F[x^{1/2},x^{1/4},x^{1/8},...]/(x)$

Notes: Local with an idempotent, nilpotent maximal ideal. Krull dimension $0$.

Keywords quotient ring polynomial ring

Reference(s):

This ring has the following properties:

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

The ring lacks the following properties:

ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
DCC annihilator (left)
DCC annihilator (right)
division ring
domain
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
nilpotent radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (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)
quasi-Frobenius
reduced
Rickart (left)
Rickart (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprimary
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly regular
T-nilpotent radical (left)
T-nilpotent radical (right)
unit regular
V ring (left)
V ring (right)
von Neumann regular

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

$\pi$-regular
Bezout (left)
Bezout (right)
cogenerator ring (left)
cogenerator ring (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
essential socle (left)
essential socle (right)
FI-injective (left)
FI-injective (right)
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)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
nonzero socle (left)
nonzero socle (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-continuous (left)
quasi-continuous (right)
self-injective (left)
self-injective (right)
serial (left)
serial (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
strongly $\pi$-regular
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)