Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.isPrime_map_of_liesOver
∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (p : Ideal R)
[inst_3 : p.IsPrime] (Sₚ : Type u_5) [inst_4 : CommSemiring Sₚ] [inst_5 : Algebra S Sₚ]
[IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] (P : Ideal S) [P.IsPrime] [P.LiesOver p],
(Ideal.map (algebraMap S Sₚ) P).IsPrime- Cited by
- 5 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement · cited by 4,706
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.mapstatement · cited by 692
- IsLocalizationstatement and proof · cited by 636
- Ideal.primeComplstatement and proof · cited by 462
- Ideal.LiesOverstatement and proof · cited by 272
- Algebra.algebraMapSubmonoidstatement and proof · cited by 137
- IsLocalization.isPrime_of_isPrime_disjointproof · cited by 13
Cited by5
Results whose statement or proof uses this declaration.
- IsLocalization.AtPrime.liesOver_map_of_liesOverproof · cited by 2
- IsLocalization.AtPrime.ramificationIdx_map_eq_ramificationIdxproof · cited by 1
- IsLocalization.AtPrime.under_map_of_isMaximalproof · cited by 1
- IsLocalization.AtPrime.equivQuotientMapOfIsMaximal_symm_apply_mkproof · cited by 1
- IsLocalization.AtPrime.exists_algebraMap_quot_eq_of_mem_quotproof · cited by 0