Description: Let $k$ be any simple ring of characteristic $0$, and let $d$ be a non-inner derivation on $k$. the differential polynomial ring $R=k[x;d]$.

Notes:

Keywords differential polynomial ring

Reference(s):

This ring has the following properties:

The ring lacks the following properties:

Artinian (left)
Artinian (right)
commutative
division ring
finite
finitely cogenerated (left)
finitely cogenerated (right)
Frobenius
local
perfect (left)
perfect (right)
primary
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
semilocal
semiperfect
semiprimary
semisimple
serial (left)
serial (right)
simple Artinian
top simple Artinian
valuation ring (left)
valuation ring (right)

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

$\pi$-regular
$I_0$
2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Baer
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
clean
cogenerator ring (left)
cogenerator ring (right)
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (left)
distributive (right)
domain
dual (left)
dual (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite uniform dimension (left)
finite uniform dimension (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)
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
NI (nilpotents form an ideal)
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
nonzero socle (right)
Ore domain (left)
Ore domain (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semiregular
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
strongly regular
symmetric
top regular
top simple
uniform (left)
uniform (right)
unit regular
V ring (left)
V ring (right)
von Neumann regular
weakly clean
Zorn