Theorems · Theorem · commutative algebra
Ideal.exists_ideal_comap_le_prime
∀ {R : Type u} {F : Type u_1} [inst : Semiring R] {S : Type u_2} [inst_1 : CommSemiring S] [inst_2 : FunLike F R S]
[inst_3 : RingHomClass F R S] {f : F} (P : Ideal R) [P.IsPrime] (I : Ideal S),
Ideal.comap f I ≤ P → ∃ Q ≥ I, Q.IsPrime ∧ Ideal.comap f Q ≤ P- Defined in
- Mathlib.RingTheory.Ideal.Maps
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coeproof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- FunLikestatement and proof · cited by 2,560
- Disjointproof · cited by 2,201
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplproof · cited by 462
- Ideal.comapstatement and proof · cited by 443
- RingHomClassstatement and proof · cited by 193
- Submonoid.mapproof · cited by 190
Cited by3
Results whose statement or proof uses this declaration.
- PrimeSpectrum.isClosed_image_of_stableUnderSpecializationproof · cited by 6
- Ideal.exists_comap_eq_of_mem_minimalPrimesproof · cited by 4
- Ideal.exists_ideal_over_prime_of_isIntegralproof · cited by 3