Prime elements in rings

This file contains lemmas about prime elements of commutative rings.

theorem mul_eq_mul_prime_prod {R : Type u_1} [integral_domain R] {α : Type u_2} [decidable_eq α] {x y a : R} {s : finset α} {p : α → R} (hp : ∀ (i : α), i ∈ s → prime (p i)) (hx : x * y = a * ∏ (i : α) in s, p i) :

∃ (t u : finset α) (b c : R), t ∪ u = s ∧ disjoint t u ∧ a = b * c ∧ x = b * ∏ (i : α) in t, p i ∧ y = c * ∏ (i : α) in u, p i

If x * y = a * ∏ i in s, p i where p i is always prime, then x and y can both be written as a divisor of a multiplied by a product over a subset of s

source

theorem mul_eq_mul_prime_pow {R : Type u_1} [integral_domain R] {x y a p : R} {n : ℕ} (hp : prime p) (hx : x * y = a * p ^ n) :

∃ (i j : ℕ) (b c : R), i + j = n ∧ a = b * c ∧ x = b * p ^ i ∧ y = c * p ^ j

If x * y = a * p ^ n where p is prime, then x and y can both be written as the product of a power of p and a divisor of a.