Description: Let $M$ be the direct sum of $\mathbb Z/p_i\mathbb Z$ where $p_i$ is the $i$'th prime. The ring $R$ is the trivial extension of $M$ by $\mathbb Z$: $\mathbb Z\times M$ with the multiplication operation $(n,a)(m,b):=(nm,nb+ma)$

Notes: Krull dimension $1$.

Keywords trivial extension

Reference(s):

The ring lacks the following properties:

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

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