Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.exists_algebraMap_quot_eq_of_mem_quot
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
[inst_3 : p.IsPrime] (Sₚ : Type u_4) [inst_4 : CommRing Sₚ] [inst_5 : Algebra S Sₚ]
[IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] (P : Ideal S) [hPp : P.LiesOver p] [P.IsMaximal]
(x : Sₚ ⧸ Ideal.map (algebraMap S Sₚ) P), ∃ a, (algebraMap S (Sₚ ⧸ Ideal.map (algebraMap S Sₚ) P)) a = x- Cited by
- 0 results in Mathlib
- Foundations
- Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites42
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Semiringproof · cited by 13,802
- 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
- IsScalarTowerproof · cited by 3,896
- HasQuotient.Quotientstatement and proof · cited by 2,301
- mul_commproof · cited by 2,262
- map_mulproof · cited by 1,137
- Ideal.IsPrimestatement and proof · cited by 827
Cited by1
Results whose statement or proof uses this declaration.
- IsLocalization.AtPrime.equivQuotientMapOfIsMaximalproof · cited by 5