Description: $\mathbb Z=\{...-3, -2, -1, 0, 1, 2, 3,...\}$ OR equivalence relation on $\mathbb N\times \mathbb N$ given by $(a,b)\sim(c,d)$ iff $a-b=c-d$

Notes:

Keywords equivalence relation

Reference(s):

This ring has the following properties:

ACC principal
atomic domain
Bezout domain
Bezout ring
coherent
connected
Dedekind domain
distributive
domain
GCD domain
Jacobson (Hilbert)
Krull domain
Mori domain
Noetherian
normal
normal domain
principal ideal domain
principal ideal ring
Prufer domain
rad-nil
reduced
Schreier domain
semiprimitive
unique factorization domain

The ring lacks the following properties:

algebraically closed field
Artinian
characteristic 0 field
dual
Euclidean field
field
finite
Frobenius
local
ordered field
perfect
perfect field
pseudo Frobenius
Pythagorean field
quadratically closed field
self-injective
semilocal
semiperfect
semiprimary
semisimple
serial
valuation
von Neumann regular

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