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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} [CancelCommMonoidWithZero R] {α : Type u_2} [DecidableEq α] {x : R} {y : R} {a : R} {s : Finset α} {p : αR} (hp : is, Prime (p i)) (hx : x * y = a * is, p i) :
∃ (t : Finset α) (u : Finset α) (b : R) (c : R), t u = s Disjoint t u a = b * c x = b * it, p i y = c * iu, p i

If x * y = a * ∏ i ∈ 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} [CancelCommMonoidWithZero R] {x : R} {y : R} {a : R} {p : R} {n : } (hp : Prime p) (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} [CommRing α] {p : α} (hp : Prime p) :
Prime (-p)
theorem Prime.abs {α : Type u_1} [CommRing α] [LinearOrder α] {p : α} (hp : Prime p) :
Prime |p|