Number theory, geometry and algebra - Dynamics-approx.jku.at

1.3 The ring Z m , residue classes
Z with the operation of addition ist an abelian group and with the two
operations of addition and multiplication it is a commutative ring with unit.
Proposition 7 A subset A ⊂ R is a subgroup of (Z,+) ⇔ A is an ideal in
the ring (Z,+,.). We then have
∨
m∈N 0
A = mZ,
(i.e. A is a principal ideal).
Proof. ⇐ is clear.
⇒: If x ∈ A, n ∈ N 0 , then
nx = (x+... +x) ∈ A ((x+... +x) n-mal)
For n ∈ {−1,−2,...} we have nx = −(−nx) ∈ A.
For the second part we can assume that A ≠ {0}. Let m be the smallest
positive element of A. We show that A = mZ. First of all mZ ⊂ A. If x ∈ A
and we write
x = qd+d 1 ,
with 0 ≤ d 1 < d (the division algorithm). Then: d 1 = x − qd ∈ A, and so
d 1 = 0.
This implies that the only quotient groups of (Z,+) resp. quotient rings
of (Z,+,.) are Z m (m ∈ N 0 ), where
Z m = Z/mZ
(i.e. Z| ∼ , where x ∼ y ⇔ m|x−y.) We write x = y (modm), if m|x−y.
The following properties are then evident:
a = b (modm) and t|m ⇒ a = b (mod|t|),
a = b (modm), r ∈ Z ⇒ ra = rb (modm),
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