Modular arithmetic
Residue class rings
The Gaussian integers.
In Z/6Z, the element 2 + 6Z is a multiple of 4 + 6Z:
If f 
Q[X] is irreducible,
then any nonzero polynomial g
R[X] of degree less than the degree of f has
an invertible residue class in R[X]/(f), and so
g + (f) divides 1.
Q[X] is irreducible,
then any nonzero polynomial g
R[X] of degree less than the degree of f has
an invertible residue class in R[X]/(f), and so
g + (f) divides 1. In Z + Zi, the element 1 + i is a divisor of 2 since (1 + i)(1 - i) = 2.
The ring
Z + Zi is a domain:
if (a + bi)(c + di) = 0,
multiply with
(a - bi)(c - di), to obtain
(a2 + b2)(c2 + d2) = 0,
from which it is clear that a = b = 0
or c = d = 0, i.e., a + bi = 0 or
c + di = 0.