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 direct product

Reference(s):

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

This ring has the following properties:

The ring lacks the following properties:

ACC principal
algebraically closed field
Artinian
atomic domain
Bezout domain
characteristic 0 field
connected
Dedekind domain
domain
dual
Euclidean domain
Euclidean field
field
finite
Frobenius
GCD domain
Krull domain
local
Mori domain
Noetherian
normal domain
ordered field
perfect
perfect field
principal ideal domain
principal ideal ring
Prufer domain
pseudo Frobenius
Pythagorean field
quadratically closed field
regular local
Schreier domain
semilocal
semiperfect
semiprimary
semisimple
serial
unique factorization domain
valuation

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