*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:

The ring lacks the following properties:

algebraically closed field
Artinian
atomic domain
Bezout domain
Bezout ring
characteristic 0 field
coherent
Dedekind domain
domain
dual
Euclidean domain
Euclidean field
field
finite
finitely pseudo Frobenius
Frobenius
GCD domain
Krull domain
Mori domain
Noetherian
normal
normal domain
ordered field
perfect field
principal ideal domain
principal ideal ring
Prufer domain
pseudo Frobenius
Pythagorean field
quadratically closed field
reduced
regular local
Schreier domain
self-injective
semiprimitive
semisimple
serial
unique factorization domain
valuation
von Neumann regular

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