# Documentation

Mathlib.RingTheory.Prime

# Prime elements in rings #

This file contains lemmas about prime elements of commutative rings.

theorem mul_eq_mul_prime_prod {R : Type u_1} {α : Type u_2} [] {x : R} {y : R} {a : R} {s : } {p : αR} (hp : is, Prime (p i)) (hx : x * y = a * Finset.prod s fun (i : α) => p i) :
∃ (t : ) (u : ) (b : R) (c : R), t u = s Disjoint t u a = b * c (x = b * Finset.prod t fun (i : α) => p i) y = c * Finset.prod u fun (i : α) => 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

theorem mul_eq_mul_prime_pow {R : Type u_1} {x : R} {y : R} {a : R} {p : R} {n : } (hp : ) (hx : x * y = a * p ^ n) :
∃ (i : ) (j : ) (b : R) (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.

theorem Prime.neg {α : Type u_1} [] {p : α} (hp : ) :
Prime (-p)
theorem Prime.abs {α : Type u_1} [] [] {p : α} (hp : ) :
Prime |p|