Description: Let $F$ be the field of two elements, and consider a countably infinite direct sum of copies of $F$. This is a countable boolean ring (without unity). After adjoining a unit element, it is still countable.

Notes:

Keywords subring direct product

Reference(s):

- (None), Publication Needed, No Citation Yet Exists: Please Add One., (1805). Citation Needed

This ring has the following properties:

$\pi$-regular
$I_0$
2-primal
Abelian
Bezout (left)
Bezout (right)
clean
coherent (left)
coherent (right)
cohopfian (left)
cohopfian (right)
commutative
Dedekind finite
distributive (left)
distributive (right)
duo (left)
duo (right)
exchange
fully semiprime
IBN
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
nonsingular (left)
nonsingular (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principally injective (left)
principally injective (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
semiregular
stable range 1
stably finite
strongly $\pi$-regular
strongly regular
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
unit regular
V ring (left)
V ring (right)
von Neumann regular
weakly clean
Zorn

The ring lacks the following properties:

ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
connected
DCC annihilator (left)
DCC annihilator (right)
division ring
domain
finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
Goldie (left)
Goldie (right)
local
Noetherian (left)
Noetherian (right)
Ore domain (left)
Ore domain (right)
orthogonally finite
perfect (left)
perfect (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
semi free ideal ring
semilocal
semiperfect
semiprimary
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly connected
top simple
top simple Artinian
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)

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

Baer
cogenerator ring (left)
cogenerator ring (right)
continuous (left)
continuous (right)
CS (left)
CS (right)
dual (left)
dual (right)
essential socle (left)
essential socle (right)
FI-injective (left)
FI-injective (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
nonzero socle (left)
nonzero socle (right)
quasi-continuous (left)
quasi-continuous (right)
self-injective (left)
self-injective (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)