Theorems · Theorem · commutative algebra
IsLocalization.under_map_of_isPrime_disjoint
∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) (S : Type u_2) [inst_1 : CommSemiring S]
[inst_2 : Algebra R S] [IsLocalization M S] {I : Ideal R},
I.IsPrime → Disjoint ↑M ↑I → Ideal.under R (Ideal.map (algebraMap R S) I) = I- Defined in
- Mathlib.RingTheory.Localization.Ideal
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 81 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.
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- SetLike.coestatement and proof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement · cited by 4,706
- Submonoidstatement and proof · cited by 3,086
- Disjointstatement and proof · cited by 2,201
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.mapstatement · cited by 692
- IsLocalizationstatement and proof · cited by 636
Cited by15
Results whose statement or proof uses this declaration.
- IsLocalization.isPrime_of_isPrime_disjointproof · cited by 13
- AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mk_mem_carrierproof · cited by 5
- IsLocalization.minimalPrimes_mapproof · cited by 4
- Ideal.under_map_of_isLocalizationAtPrimeproof · cited by 3
- IsLocalization.isMaximal_iff_isMaximal_disjointproof · cited by 3
- IsLocalization.liesOver_map_of_isPrime_disjointproof · cited by 2
- PrimeSpectrum.localization_comap_rangeproof · cited by 2
- IsLocalization.liesOver_of_isPrime_of_disjointproof · cited by 1
- isJacobsonRing_localizationproof · cited by 1
- IsLocalization.isMaximal_of_isMaximal_disjointproof · cited by 1
- Polynomial.height_eq_height_add_oneproof · cited by 1
- Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOverproof · cited by 1