Description: The ring is the trivial extension $T(\mathbb Z, \mathbb Z_{2^\infty})$ of the Prüfer $2$-group.

Notes: Krull dimension $1$.

Keywords trivial extension

Reference(s):

This ring has the following properties:

2-primal
Abelian
commutative
connected
CS (left)
CS (right)
Dedekind finite
duo (left)
duo (right)
essential socle (left)
essential socle (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reversible
semicommutative (SI condition, zero-insertive)
simple socle (left)
simple socle (right)
stably finite
strongly connected
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
uniform (left)
uniform (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
clean
cogenerator ring (left)
cogenerator ring (right)
continuous (left)
continuous (right)
division ring
domain
dual (left)
dual (right)
exchange
FI-injective (left)
FI-injective (right)
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
hereditary (left)
hereditary (right)
Kasch (left)
Kasch (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)
principally injective (left)
principally injective (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
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular
Zorn

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

ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Bezout (left)
Bezout (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
DCC annihilator (left)
DCC annihilator (right)
distributive (left)
distributive (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Goldie (left)
Goldie (right)
simple-injective (left)
simple-injective (right)
stable range 1
weakly clean