Theorems · Theorem · commutative algebra
Ideal.dvd_span_singleton
∀ {A : Type u_2} [inst : CommRing A] [IsDedekindDomain A] {I : Ideal A} {x : A}, I ∣ Ideal.span {x} ↔ x ∈ I- Cited by
- 5 results in Mathlib
- Foundations
- Depth 145 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.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Ideal.spanstatement · cited by 948
- IsDedekindDomainstatement and proof · cited by 668
- Set.singleton_subset_iffproof · cited by 206
- Ideal.span_leproof · cited by 70
- Ideal.dvd_iff_leproof · cited by 33
Cited by5
Results whose statement or proof uses this declaration.
- IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvdproof · cited by 5
- IsDedekindDomain.HeightOneSpectrum.intValuation_exists_uniformizerproof · cited by 3
- IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_memproof · cited by 3
- IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_memproof · cited by 1
- IsCyclotomicExtension.Rat.isCoprime_of_not_zeta_sub_one_dvdproof · cited by 0