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:

2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Baer
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
coherent (left)
coherent (right)
commutative
connected
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (left)
distributive (right)
domain
duo (left)
duo (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
free ideal ring (left)
free ideal ring (right)
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
prime
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semiprime
semiprimitive
stably finite
strongly connected
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
uniform (left)
uniform (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
Artinian (left)
Artinian (right)
clean
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
continuous (left)
continuous (right)
division ring
dual (left)
dual (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite
finitely cogenerated (left)
finitely cogenerated (right)
Frobenius
fully prime
fully semiprime
Kasch (left)
Kasch (right)
local
nonzero socle (left)
nonzero socle (right)
perfect (left)
perfect (right)
primary
primitive (left)
primitive (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
semilocal
semiperfect
semiprimary
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
simple socle (left)
simple socle (right)
stable range 1
strongly $\pi$-regular
strongly regular
top regular
top simple
top simple Artinian
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular
Zorn

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