Theorems · Theorem · commutative algebra
Ideal.eq_bot_of_comap_eq_bot
∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] {I : Ideal S} [inst_2 : Algebra R S]
[Nontrivial R] [IsDomain S] [Algebra.IsIntegral R S], Ideal.comap (algebraMap R S) I = ⊥ → I = ⊥- Defined in
- Mathlib.RingTheory.Ideal.GoingUp
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Algebra.algebraMapstatement and proof · cited by 4,706
- Nontrivialstatement and proof · cited by 2,416
- IsDomainstatement and proof · cited by 2,196
- Ideal.comapstatement and proof · cited by 443
- Algebra.IsIntegralstatement and proof · cited by 224
- eq_bot_iffproof · cited by 159
- Algebra.IsIntegral.isIntegralproof · cited by 86
Cited by9
Results whose statement or proof uses this declaration.
- Ideal.eq_bot_of_liesOver_botproof · cited by 5
- Ideal.under_ne_botproof · cited by 2
- isJacobsonRing_of_isIntegralproof · cited by 1
- Ideal.primesOver_botproof · cited by 1
- IsLocalization.OverPrime.mem_normalizedFactors_of_isPrimeproof · cited by 1
- not_dvd_differentIdeal_iffproof · cited by 1
- NumberField.exists_not_isUnramifiedAt_int_of_isGaloisproof · cited by 0
- Nat.absNorm_under_primeproof · cited by 0