Description: Let $D$ be a countable, existentially closed division ring over a field $F$. Let $S=D\otimes_F F(x)$. Let $R$ be the trivial extension $T(S,D)$ of $S$ by the $S$-module $D$$

Notes:

Keywords rational polynomial ring tensor product trivial extension

Reference(s):

- Nicholson, W. Keith; Mohamed F. Yousif., Quasi-Frobenius Rings., Vol. 158. Cambridge University Press, (2003). Example 8.16 Pg 212

This ring has the following properties:

The ring lacks the following properties:

$\pi$-regular
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
division ring
domain
dual (left)
FI-injective (right)
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
principal ideal domain (left)
principal ideal domain (right)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
semi free ideal ring
semiprimary
semisimple
simple Artinian
strongly $\pi$-regular
strongly regular
unit regular
von Neumann regular

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

$I_0$
2-primal
Abelian
ACC annihilator (left)
ACC principal (left)
Baer
Bezout (left)
Bezout (right)
clean
cogenerator ring (right)
coherent (left)
cohopfian (left)
cohopfian (right)
commutative
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (right)
distributive (left)
distributive (right)
duo (left)
duo (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
finite uniform dimension (left)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
fully prime
fully semiprime
Goldie (left)
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
lift/rad
local
NI (nilpotents form an ideal)
nil radical
nilpotent radical
Noetherian (left)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
Ore ring (left)
Ore ring (right)
polynomial identity
prime
primitive (left)
primitive (right)
principal ideal ring (left)
principal ideal ring (right)
principally injective (left)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprime
semiprimitive
semiregular
serial (left)
serial (right)
simple
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
strongly connected
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
weakly clean
Zorn