Theorems · Theorem · commutative algebra
Ideal.eq_bot_of_liesOver_bot
∀ (A : Type u_1) [inst : CommRing A] {B : Type u_2} [inst_1 : CommRing B] [inst_2 : Algebra A B]
[Algebra.IsIntegral A B] (P : Ideal B) [Nontrivial A] [IsDomain B] [h : P.LiesOver ⊥], P = ⊥- Defined in
- Mathlib.RingTheory.Ideal.GoingUp
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Bot.botstatement and proof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- IsDomainstatement and proof · cited by 2,196
- Ideal.LiesOverstatement and proof · cited by 272
- Algebra.IsIntegralstatement and proof · cited by 224
- Ideal.liesOver_iffproof · cited by 10
- Ideal.eq_bot_of_comap_eq_botproof · cited by 9
Cited by5
Results whose statement or proof uses this declaration.
- Ideal.absNorm_pow_inertiaDegproof · cited by 2
- Ideal.exists_relNorm_eq_pow_of_isPrimeproof · cited by 1
- Ideal.inertiaDeg'_botproof · cited by 1
- Ideal.relNorm_eq_pow_of_isPrime_isGaloisproof · cited by 1
- Algebra.isUnramifiedIn_botproof · cited by 0