Theorems · Theorem · commutative algebra
IsRadical.dvd_radical
∀ {M : Type u_1} [inst : CommMonoidWithZero M] [inst_1 : NormalizationMonoid M] [inst_2 : UniqueFactorizationMonoid M]
{a : M}, IsRadical a → a ≠ 0 → a ∣ UniqueFactorizationMonoid.radical aIf a is a radical element, then it divides its radical.
- Defined in
- Mathlib.RingTheory.Radical.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommMonoidWithZerostatement and proof · cited by 913
- UniqueFactorizationMonoidstatement and proof · cited by 279
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.radicalstatement · cited by 64
- IsRadicalstatement and proof · cited by 16
- Associated.dvd'proof · cited by 7
- UniqueFactorizationMonoid.radical_associatedproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.dvd_radical_iffproof · cited by 1