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
2-primal
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
clean
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
I_0
IBN
lift/rad
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonzero socle (left)
nonzero socle (right)
orthogonally finite
perfect (left)
perfect (right)
quasi-duo (left)
quasi-duo (right)
reversible
semilocal
semiperfect
semiprimary
semiregular
stable range 1
stably finite
strongly $\pi$-regular
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
Zorn

The ring lacks the following properties:

Bezout domain (left)
Bezout domain (right)
commutative
division ring
domain
free ideal ring (left)
free ideal ring (right)
fully prime
fully semiprime
Ore domain (left)
Ore domain (right)
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
reduced
semi free ideal ring
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:

Abelian
Baer
Bezout (left)
Bezout (right)
cogenerator ring (left)
cogenerator ring (right)
connected
continuous (left)
continuous (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
FI-injective (left)
FI-injective (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Frobenius
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
local
nonsingular (left)
nonsingular (right)
Ore ring (left)
Ore ring (right)
polynomial identity
primary
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semihereditary (left)
semihereditary (right)
serial (left)
serial (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
top simple
top simple Artinian
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)