Theorems · Theorem · commutative algebra
Ideal.mem_normalizedFactors_iff
∀ {A : Type u_2} [inst : CommRing A] [inst_1 : IsDedekindDomain A] {p I : Ideal A},
I ≠ ⊥ → (p ∈ UniqueFactorizationMonoid.normalizedFactors I ↔ p.IsPrime ∧ I ≤ p)- Cited by
- 3 results in Mathlib
- Foundations
- Depth 149 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsDedekindDomain
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Multisetstatement · cited by 2,627
- Ideal.IsPrimestatement and proof · cited by 827
- IsDedekindDomainstatement and proof · cited by 668
- UniqueFactorizationMonoid.normalizedFactorsstatement and proof · cited by 151
- Ideal.dvd_iff_leproof · cited by 33
- Ideal.zero_eq_botproof · cited by 27
- Ideal.prime_iff_isPrimeproof · cited by 6
- UniqueFactorizationMonoid.mem_normalizedFactors_iffproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Ideal.absNorm_relNormproof · cited by 2
- Ideal.mem_primesOver_iff_mem_normalizedFactorsproof · cited by 2
- RingOfIntegers.isPrincipalIdealRing_of_isPrincipal_of_norm_le_of_isPrimeproof · cited by 1