User:Virginia-American/Sandbox/Bezout's lemma
Proof[edit]
Bezout's lemma can be proved as a corollary of the proof that the integers are a PID.[1]
Modules[edit]
Definition: A ideal M is a set of numbers closed under addition and subtraction.[2] 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.