Theorems · Theorem · commutative algebra
Ideal.exists_ideal_le_liesOver_of_le
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.HasGoingDown R S] {p q : Ideal R} [p.IsPrime] [q.IsPrime] (Q : Ideal S) [Q.IsPrime] [Q.LiesOver q],
p ≤ q → ∃ P ≤ Q, P.IsPrime ∧ P.LiesOver p- Defined in
- Mathlib.RingTheory.Ideal.GoingDown
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 25 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.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Idealstatement and proof · cited by 4,748
- le_reflproof · cited by 2,061
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.LiesOverstatement and proof · cited by 272
- lt_of_le_of_neproof · cited by 230
- Ideal.underproof · cited by 170
- Ideal.over_defproof · cited by 60
- Algebra.HasGoingDownstatement and proof · cited by 11
- Algebra.HasGoingDown.exists_ideal_le_liesOver_of_ltproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Algebra.HasGoingDown.iff_generalizingMap_primeSpectrumComapproof · cited by 4
- Ideal.exists_ideal_lt_liesOver_of_ltproof · cited by 1