Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.radical_dvd_radical_iff_normalizedFactors_subset_normalizedFactors
∀ {M : Type u_1} [inst : CommMonoidWithZero M] [inst_1 : NormalizationMonoid M] [inst_2 : UniqueFactorizationMonoid M]
{a b : M},
UniqueFactorizationMonoid.radical a ∣ UniqueFactorizationMonoid.radical b ↔
UniqueFactorizationMonoid.normalizedFactors a ⊆ UniqueFactorizationMonoid.normalizedFactors b- Defined in
- Mathlib.RingTheory.Radical.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Multisetstatement · cited by 2,627
- Nontrivialproof · cited by 2,416
- eq_or_neproof · cited by 1,117
- CommMonoidWithZerostatement and proof · cited by 913
- UniqueFactorizationMonoidstatement and proof · cited by 279
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.normalizedFactorsstatement and proof · cited by 151
- UniqueFactorizationMonoid.radicalstatement and proof · cited by 64
- UniqueFactorizationMonoid.primeFactorsproof · cited by 32
- UniqueFactorizationMonoid.normalizedFactors_zeroproof · cited by 14
- UniqueFactorizationMonoid.dvd_iff_normalizedFactors_le_normalizedFactorsproof · cited by 11
- Multiset.le_iff_subsetproof · cited by 7
Cited by2
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.radical_dvd_radicalproof · cited by 2