Theorems · Theorem · commutative algebra
Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomain
- 1000+ list: Going-up and going-down theorems
∀ {R : Type u_1} [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] [inst_2 : Algebra R S] [IsDomain S]
[Algebra.IsIntegral R S] (P : Ideal R) [P.IsPrime],
RingHom.ker (algebraMap R S) ≤ P → ∃ Q, Q.IsPrime ∧ Ideal.comap (algebraMap R S) Q = Pcomap (algebraMap R S) is a surjection from the prime spec of R to prime spec of S.
hP : (algebraMap R S).ker ≤ P is a slight generalization of the extension being injective
- Defined in
- Mathlib.RingTheory.Ideal.GoingUp
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites31
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement and proof · cited by 10,189
- SetLike.coeproof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- RingHom.compproof · cited by 899
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplproof · cited by 462
- Ideal.IsMaximalproof · cited by 452
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.exists_ideal_over_prime_of_isIntegral_of_isPrimeproof · cited by 2
- Algebra.IsIntegral.nontrivial_heightOneSpectrumproof · cited by 0