User:Virginia-American/Sandbox/Bezout's lemma

Proof
Bezout's lemma can be proved as a corollary of the proof that the integers are a PID.

Modules
Definition: A ideal M is a set of numbers closed under addition and subtraction. In symbols, if a, b ∈ M then a ± b ∈ M.

Lemma: If M is a ideal, 0 ∈ M. Proof: let a ∈ M. Then a − a = 0 ∈ M.

Definition: The set M = {0} is called the zero ideal.

Definition: A ideal that contains a number other than 0 is called a nonzero ideal.

Lemma: If M is a nonzero ideal it contains a postiive number. Proof: let a ∈ M, a ≠ 0. Either a > 0 or M ∋ 0 − a > 0.

Lemma: The set of all multiples of a number d, M = {..., −2d, −d, 0, d, 2d,...} is an ideal. Proof: Let a = md, b = nd ∈ M. Then a ± b = (m ± n)d ∈ M.