Theorems · Theorem · commutative algebra
Ideal.exists_le_prime_disjoint
∀ {α : Type u} [inst : CommSemiring α] (I : Ideal α) (S : Submonoid α),
Disjoint ↑I ↑S → ∃ p, p.IsPrime ∧ I ≤ p ∧ Disjoint ↑p ↑S- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coestatement and proof · cited by 8,199
- Set.Elemproof · cited by 7,166
- Set.ofPredproof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- Submonoidstatement and proof · cited by 3,086
- Disjointstatement and proof · cited by 2,201
- SupSet.sSupproof · cited by 954
- Ideal.IsPrimestatement and proof · cited by 827
- IsEmptyproof · cited by 759
- isEmpty_or_nonemptyproof · cited by 269
Cited by5
Results whose statement or proof uses this declaration.
- Ideal.exists_notMem_dvd_algebraMap_of_primesOver_eq_singletonproof · cited by 3
- Ideal.exists_ideal_comap_le_primeproof · cited by 3
- Ideal.comap_map_eq_self_iff_of_isPrimeproof · cited by 2
- Ideal.exists_le_prime_notMem_of_isIdempotentElemproof · cited by 1
- PrimeSpectrum.isLocalization_away_iff_atPrime_of_basicOpen_eq_singletonproof · cited by 1