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:

The ring lacks the following properties:

ACC principal
algebraically closed field
Artinian
atomic domain
Bezout domain
characteristic 0 field
Dedekind domain
domain
Euclidean domain
Euclidean field
field
Frobenius
GCD domain
Krull domain
Mori domain
Noetherian
normal
normal domain
ordered field
perfect field
principal ideal domain
principal ideal ring
Prufer domain
Pythagorean field
quadratically closed field
reduced
regular local
Schreier domain
semiprimitive
semisimple
unique factorization domain
von Neumann regular

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

Bezout ring
clean
Cohen-Macaulay
coherent
continuous
distributive
dual
finite
finitely pseudo Frobenius
Gorenstein
Jacobson (Hilbert)
Kasch
local complete intersection ring
perfect
pseudo Frobenius
rad-nil
regular
self-injective
semiperfect
semiprimary
semiregular
serial
strongly pi regular
valuation