Description: The quotient of the real polynomial ring $\mathbb R[x]$ by the ideal generated by $x^2$

Notes: Krull dimension 0

Keywords quotient ring polynomial ring

Reference(s):

This ring has the following properties:

ACC principal
Artinian
Bezout ring
clean
Cohen-Macaulay
coherent
connected
continuous
dual
finitely pseudo Frobenius
Frobenius
Gorenstein
local
Noetherian
perfect
principal ideal ring
pseudo Frobenius
self-injective
semilocal
semiperfect
semiprimary
semiregular
serial
stable range 1
strongly pi regular
valuation

The ring lacks the following properties:

algebraically closed field
atomic domain
Bezout domain
characteristic 0 field
Dedekind domain
domain
Euclidean domain
Euclidean field
field
finite
GCD domain
Krull domain
Mori domain
normal
normal domain
ordered field
perfect field
principal ideal domain
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: