Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.irreducible_iff_prime
∀ {α : Type u_2} {inst : CommMonoidWithZero α} [self : UniqueFactorizationMonoid α] {a : α}, Irreducible a ↔ Prime a- Cited by
- 8 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- Assumes
- UniqueFactorizationMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommMonoidWithZerostatement and proof · cited by 913
- Irreduciblestatement · cited by 496
- UniqueFactorizationMonoidstatement and proof · cited by 279
- Primestatement · cited by 277
Cited by8
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.factors_uniqueproof · cited by 14
- IsDedekindDomain.HeightOneSpectrum.irreducibleproof · cited by 13
- Associates.count_of_coprimeproof · cited by 3
- UniqueFactorizationMonoid.isRelPrime_iff_no_prime_factorsproof · cited by 2
- Associates.exists_prime_dvd_of_not_inf_oneproof · cited by 1
- Ideal.isPrime_of_irreducible_absNormproof · cited by 1
- IsDiscreteValuationRing.of_ufd_of_unique_irreducibleproof · cited by 1
- UniqueFactorizationMonoid.exists_prime_iffproof · cited by 0