Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.to_map_mem_maximal_iff
∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (I : Ideal R)
[hI : I.IsPrime] [inst_3 : IsLocalization.AtPrime S I] (x : R) (h : optParam (IsLocalRing S) ⋯),
(algebraMap R S) x ∈ IsLocalRing.maximalIdeal S ↔ x ∈ I- Cited by
- 9 results in Mathlib
- Foundations
- Depth 84 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.
- DFunLike.coestatement and proof · cited by 62,936
- 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 and proof · cited by 4,706
- Ideal.IsPrimestatement and proof · cited by 827
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement · cited by 297
- not_iff_notproof · cited by 159
- IsLocalization.AtPrimestatement and proof · cited by 79
- IsLocalization.AtPrime.isLocalRingstatement · cited by 19
Cited by9
Results whose statement or proof uses this declaration.
- IsLocalization.AtPrime.under_maximalIdealproof · cited by 7
- Ideal.algebraMap_residueField_eq_zeroproof · cited by 5
- IsLocalization.AtPrime.isUnit_mk'_iffproof · cited by 3
- IsLocalization.AtPrime.not_isFieldproof · cited by 2
- IsArtinianRing.exists_not_mem_forall_mem_of_neproof · cited by 1
- IsLocalization.OverPrime.mem_normalizedFactors_of_isPrimeproof · cited by 1
- ValuationSubring.idealOfLE_ofPrimeproof · cited by 0
- AlgebraicGeometry.ΓSpec.right_triangleproof · cited by 0