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: Krull dimension $1$.

Keywords equivalence relation

Reference(s):

This ring has the following properties:

ACC principal
atomic domain
Bezout domain
Bezout ring
Cohen-Macaulay
coherent
connected
Dedekind domain
distributive
domain
GCD domain
Gorenstein
Jacobson (Hilbert)
Krull domain
Mori domain
Noetherian
normal
normal domain
principal ideal domain
principal ideal ring
Prufer domain
rad-nil
reduced
regular
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
regular local
self-injective
semilocal
semiperfect
semiprimary
semisimple
serial
valuation
von Neumann regular

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