Description: Let $s:F\to F$ be a field automorphism such that $[F:s(F)]>1$. $F[x,s]$ is the skew polynomial ring where $xa:=s(a)x$ for all $a$ in $F$. The ring is $F[x;s]/(x^2)$.

Notes: Composition length different on both sides

Keywords quotient ring skew polynomial 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)
Bezout (left)
clean
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
connected
CS (left)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (left)
dual (left)
duo (left)
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
Ikeda-Nakayama (left)
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
principal ideal ring (left)
quasi-continuous (left)
quasi-duo (left)
quasi-duo (right)
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semiregular
serial (left)
simple socle (left)
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
uniform (left)
valuation ring (left)
weakly clean
Zorn

The ring lacks the following properties:

Baer
Bezout (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
commutative
distributive (right)
division ring
domain
dual (right)
duo (right)
FI-injective (right)
finite
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
hereditary (left)
hereditary (right)
Ikeda-Nakayama (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (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)
semiprime
semiprimitive
semisimple
serial (right)
simple
simple Artinian
simple socle (right)
simple-injective (right)
strongly regular
uniform (right)
unit regular
V ring (left)
V ring (right)
valuation ring (right)
von Neumann regular

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