Description: Let $V$ be a countable dimensional right vector space over a division ring $D$. $R=End(V_D)$.

Notes: Has exactly three ideals: the nontrivial one is the set of transformations with finite dimensional range.

Keywords endomorphism ring ring of functions

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
Bezout (left)
Bezout (right)
clean
coherent (left)
coherent (right)
connected
continuous (right)
CS (right)
exchange
FI-injective (right)
fully prime
fully semiprime
lift/rad
nil radical
nilpotent radical
nonsingular (left)
nonsingular (right)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
prime
primitive (left)
primitive (right)
principally injective (left)
principally injective (right)
quasi-continuous (right)
Rickart (left)
Rickart (right)
self-injective (right)
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
semiregular
simple-injective (right)
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
von Neumann regular
weakly clean
Zorn

The ring lacks the following properties:

2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
commutative
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (left)
distributive (right)
division ring
domain
dual (left)
dual (right)
duo (left)
duo (right)
finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
Goldie (left)
Goldie (right)
hereditary (right)
IBN
Kasch (left)
Kasch (right)
local
NI (nilpotents form an ideal)
Noetherian (left)
Noetherian (right)
Ore domain (left)
Ore domain (right)
orthogonally finite
perfect (left)
perfect (right)
primary
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-duo (left)
quasi-duo (right)
quasi-Frobenius
reduced
reversible
self-injective (left)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semisimple
serial (left)
serial (right)
simple
simple Artinian
simple socle (left)
simple socle (right)
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
strongly regular
symmetric
top simple Artinian
uniform (left)
uniform (right)
unit regular
V ring (left)
valuation ring (left)
valuation ring (right)

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

Baer
continuous (left)
CS (left)
essential socle (left)
essential socle (right)
FI-injective (left)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
hereditary (left)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
polynomial identity
quasi-continuous (left)
simple-injective (left)
top simple
V ring (right)