Description: For field $k$, $k\langle x,y,z\rangle /I$ where $I=(FxF)^2+(FyF)^2+(FzF)^2 + FxyzF + FyzxF + FzxyF$

Notes: $13$ dimensional $k$ algebra.

Keywords quotient ring

Reference(s):

- Greg Marks, Reversible And Symmetric Rings, Journal Of Pure And Applied Algebra Volume 174, Issue 3, Pp 311–318, (2002). Example 5 P 315

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
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
connected
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
essential socle (left)
essential socle (right)
exchange
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)
IBN
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
quasi-duo (left)
quasi-duo (right)
reversible
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semiregular
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:

Bezout domain (left)
Bezout domain (right)
commutative
division ring
domain
duo (left)
duo (right)
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:

Baer
Bezout (left)
Bezout (right)
cogenerator ring (left)
cogenerator ring (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
distributive (left)
distributive (right)
dual (left)
dual (right)
FI-injective (left)
FI-injective (right)
finite
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
Frobenius
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
nonsingular (left)
nonsingular (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-continuous (left)
quasi-continuous (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)
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)