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):

This ring has the following properties:

2-primal
Abelian
commutative
Dedekind finite
duo (left)
duo (right)
IBN
Kasch (left)
Kasch (right)
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
polynomial identity
quasi-duo (left)
quasi-duo (right)
reversible
semicommutative (SI condition, zero-insertive)
stably finite
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)

The ring lacks the following properties:

$\pi$-regular
Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
continuous (left)
continuous (right)
division ring
domain
finite
finite uniform dimension (left)
finite uniform dimension (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
local
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)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly $\pi$-regular
strongly regular
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular

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

$I_0$
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Bezout (left)
Bezout (right)
clean
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
connected
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (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)
orthogonally finite
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
strongly connected
weakly clean
Zorn