Theorems · Theorem · commutative algebra
Ideal.exists_ideal_ge_liesOver_of_le
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.HasGoingUp R S] {p q : Ideal R} [q.IsPrime] (P : Ideal S) [P.IsPrime] [P.LiesOver p],
p ≤ q → ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q- Defined in
- Mathlib.RingTheory.Ideal.HasGoingUp
- 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
- eq_or_neproof · cited by 1,117
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.LiesOverstatement and proof · cited by 272
- lt_of_le_of_neproof · cited by 230
- Ideal.over_defproof · cited by 60
- Algebra.HasGoingUpstatement and proof · cited by 6
- Algebra.HasGoingUp.exists_ideal_ge_liesOver_of_ltproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.exists_ideal_gt_liesOver_of_ltproof · cited by 1
- Algebra.HasGoingUp.iff_specializingMap_primeSpectrumComapproof · cited by 1