Description: Group ring of the quaternion group $\mathbb Q_8$ with the field of two elements $F_2$.

Notes:

Keywords group ring

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
clean
cogenerator ring (left)
cogenerator ring (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
dual (left)
dual (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Frobenius
Goldie (left)
Goldie (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
lift/rad
local
NI (nilpotents form an ideal)
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonzero socle (left)
nonzero socle (right)
orthogonally finite
perfect (left)
perfect (right)
primary
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
quasi-Frobenius
reversible
self-injective (left)
self-injective (right)
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semiregular
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
weakly clean
Zorn

The ring lacks the following properties:

Baer
Bezout domain (left)
Bezout domain (right)
commutative
division ring
domain
free ideal ring (left)
free ideal ring (right)
fully prime
fully semiprime
hereditary (left)
hereditary (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
reduced
Rickart (left)
Rickart (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly regular
symmetric
unit regular
V ring (left)
V ring (right)
von Neumann regular

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

Bezout (left)
Bezout (right)
distributive (left)
distributive (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal ring (left)
principal ideal ring (right)
serial (left)
serial (right)
simple socle (left)
simple socle (right)
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)