Description: Twisted polynomials $k[x;s]$ for a division ring $k$ with endomorphism s which isn't an automorphism. The twist is given by $xa=xs(a)$.

Notes:

Keywords twisted polynomial ring

Reference(s):

- Lam, Tsit-Yuen., A First Course In Noncommutative Rings, Vol. 131. Springer, (2001). P 21

This ring has the following properties:

2-primal
Abelian
ACC annihilator (left)
ACC principal (left)
Baer
Bezout (left)
Bezout domain (left)
coherent (left)
coherent (right)
connected
CS (left)
DCC annihilator (right)
Dedekind finite
domain
finite uniform dimension (left)
finitely generated socle (left)
free ideal ring (left)
Goldie (left)
hereditary (left)
IBN
Ikeda-Nakayama (left)
NI (nilpotents form an ideal)
Noetherian (left)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore ring (left)
orthogonally finite
prime
principal ideal domain (left)
principal ideal ring (left)
quasi-continuous (left)
reduced
reversible
Rickart (left)
Rickart (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semiprime
stably finite
strongly connected
symmetric

The ring lacks the following properties:

$\pi$-regular
Artinian (left)
Artinian (right)
Bezout (right)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
commutative
distributive (right)
division ring
dual (left)
dual (right)
essential socle (left)
essential socle (right)
FI-injective (left)
FI-injective (right)
finite
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
Frobenius
Goldie (right)
Kasch (left)
Kasch (right)
Noetherian (right)
nonzero socle (left)
nonzero socle (right)
Ore domain (right)
Ore ring (right)
perfect (left)
perfect (right)
primary
principal ideal domain (right)
principal ideal ring (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
semiprimary
semisimple
serial (right)
simple Artinian
simple socle (left)
simple socle (right)
strongly $\pi$-regular
strongly regular
uniform (right)
unit regular
valuation ring (right)
von Neumann regular

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

$I_0$
ACC annihilator (right)
ACC principal (right)
clean
continuous (left)
continuous (right)
CS (right)
DCC annihilator (left)
distributive (left)
duo (left)
duo (right)
exchange
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (right)
fully prime
fully semiprime
hereditary (right)
Ikeda-Nakayama (right)
lift/rad
local
nil radical
nilpotent radical
polynomial identity
primitive (left)
primitive (right)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
semilocal
semiperfect
semiprimitive
semiregular
serial (left)
simple
simple-injective (left)
simple-injective (right)
stable range 1
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
uniform (left)
V ring (left)
V ring (right)
valuation ring (left)
weakly clean
Zorn