Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.radical_ne_zero
∀ {M : Type u_1} [inst : CommMonoidWithZero M] [inst_1 : NormalizationMonoid M] [inst_2 : UniqueFactorizationMonoid M]
{a : M} [Nontrivial M], UniqueFactorizationMonoid.radical a ≠ 0- Defined in
- Mathlib.RingTheory.Radical.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- Finsetproof · cited by 13,712
- Nontrivialstatement and proof · cited by 2,416
- CommMonoidWithZerostatement and proof · cited by 913
- Finset.valproof · cited by 438
- UniqueFactorizationMonoidstatement and proof · cited by 279
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.radicalstatement · cited by 64
- UniqueFactorizationMonoid.primeFactorsproof · cited by 32
- UniqueFactorizationMonoid.zero_notMem_normalizedFactorsproof · cited by 5
- Finset.prod_valproof · cited by 5
- Multiset.prod_ne_zeroproof · cited by 4
Cited by5
Results whose statement or proof uses this declaration.
- EuclideanDomain.radical_mul_divRadicalproof · cited by 5
- divRadical_dvd_derivativeproof · cited by 1
- EuclideanDomain.eq_divRadicalproof · cited by 1
- Nat.radical_posproof · cited by 1