Description: Let $k$ be a field and $X=\{s,t,u,v,w,x,y,z\}$ be indeterminates. Let $R$ be the free algebra over $k$ and $X$ modulo relations which make the matrix equation $AB=I_2$ hold, where $A=\begin{bmatrix}s&t\\ u&v\end{bmatrix}$ and $B=\begin{bmatrix}w&x\\ y&z\end{bmatrix}$.

Notes:

Keywords quotient ring

Reference(s):

This ring has the following properties:

The ring lacks the following properties:

$\pi$-regular
$I_0$
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
clean
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
commutative
continuous (left)
continuous (right)
distributive (left)
distributive (right)
division ring
dual (left)
dual (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)
Frobenius
Goldie (left)
Goldie (right)
Kasch (left)
Kasch (right)
local
Noetherian (left)
Noetherian (right)
nonzero socle (left)
nonzero socle (right)
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
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
self-injective (left)
self-injective (right)
semilocal
semiperfect
semiprimary
semiregular
semisimple
serial (left)
serial (right)
simple Artinian
simple socle (left)
simple socle (right)
stable range 1
stably finite
strongly $\pi$-regular
strongly regular
top simple Artinian
uniform (left)
uniform (right)
unit regular
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)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
duo (left)
duo (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
fully prime
fully semiprime
hereditary (left)
hereditary (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
nil radical
nilpotent radical
Ore ring (left)
Ore ring (right)
polynomial identity
primitive (left)
primitive (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprimitive
simple
simple-injective (left)
simple-injective (right)
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
V ring (left)
V ring (right)
weakly clean