Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.primeFactors_eq_empty_iff
∀ {M : Type u_1} [inst : CommMonoidWithZero M] [inst_1 : NormalizationMonoid M] [inst_2 : UniqueFactorizationMonoid M]
{a : M}, a ≠ 0 → (UniqueFactorizationMonoid.primeFactors a = ∅ ↔ IsUnit a)The finset of prime factors of x is empty if and only if x is a unit.
The converse is true without the nonzero assumption, see primeFactors_of_isUnit.
- Defined in
- Mathlib.RingTheory.Radical.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- IsUnitstatement and proof · cited by 1,602
- CommMonoidWithZerostatement and proof · cited by 913
- UniqueFactorizationMonoidstatement and proof · cited by 279
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.primeFactorsstatement · cited by 32
- Multiset.toFinset_eq_emptyproof · cited by 2
- UniqueFactorizationMonoid.normalizedFactors_eq_zero_iffproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.radical_eq_one_iffproof · cited by 0