Description: Let $S$ be the subset of infinite matrices over a field $k$ which are nonzero only on finitely many entries above the diagonal. $R$ is the subring of the matrix ring generated by $S$ and the "infinite" identity matrix.

Notes:

Keywords infinite matrix ring subring

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
Abelian
ACC principal (right)
clean
cohopfian (left)
cohopfian (right)
connected
Dedekind finite
essential socle (left)
exchange
IBN
lift/rad
local
nil radical
nonzero socle (left)
orthogonally finite
perfect (right)
quasi-duo (left)
quasi-duo (right)
semilocal
semiperfect
semiregular
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
weakly clean
Zorn

The ring lacks the following properties:

Artinian (left)
Artinian (right)
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
commutative
distributive (left)
distributive (right)
division ring
domain
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
nilpotent radical
Noetherian (left)
Noetherian (right)
Ore domain (left)
Ore domain (right)
perfect (left)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
quasi-Frobenius
semi free ideal ring
semiprimary
semiprimitive
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly regular
T-nilpotent radical (left)
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular

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

2-primal
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
Baer
cogenerator ring (left)
cogenerator ring (right)
coherent (left)
coherent (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
dual (left)
dual (right)
duo (left)
duo (right)
essential socle (right)
FI-injective (left)
FI-injective (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
NI (nilpotents form an ideal)
nonsingular (left)
nonsingular (right)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-continuous (left)
quasi-continuous (right)
reduced
reversible
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semiprime
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
symmetric
uniform (left)
uniform (right)