Theorems · Theorem · commutative algebra
Ideal.exists_ideal_over_prime_of_isIntegral
∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.IsIntegral R S] (P : Ideal R) [P.IsPrime] (I : Ideal S),
Ideal.comap (algebraMap R S) I ≤ P → ∃ Q ≥ I, Q.IsPrime ∧ Ideal.comap (algebraMap R S) Q = P- Defined in
- Mathlib.RingTheory.Ideal.GoingUp
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 118 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
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- LE.le.transproof · cited by 3,151
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.comapstatement and proof · cited by 443
- Algebra.IsIntegralstatement and proof · cited by 224
- Ideal.exists_ideal_comap_le_primeproof · cited by 3
- Ideal.exists_ideal_over_prime_of_isIntegral_of_isPrimeproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- Ideal.exists_ideal_over_maximal_of_isIntegralproof · cited by 5
- Algebra.IsIntegral.comap_surjectiveproof · cited by 2
- PrimeSpectrum.isClosedMap_comap_of_isIntegralproof · cited by 2