Theorems · Theorem · commutative algebra
IsLocalization.map_eq_zero_iff
∀ {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] (r : R), (algebraMap R S) r = 0 ↔ ∃ m, ↑m * r = 0- Defined in
- Mathlib.RingTheory.Localization.Defs
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement · cited by 4,706
- Submonoidstatement and proof · cited by 3,086
- IsLocalizationstatement and proof · cited by 636
- IsLocalization.toLocalizationMapproof · cited by 69
- Submonoid.LocalizationMap.map_eq_zero_iffproof · cited by 3
Cited by12
Results whose statement or proof uses this declaration.
- Algebra.Generators.liftBaseChange_injective_of_isLocalizationAwayproof · cited by 2
- IsLocalization.Away.exists_isIntegral_mul_of_isIntegral_algebraMapproof · cited by 2
- IsLocalization.exists_isIntegral_smul_of_isIntegral_mapproof · cited by 2
- IsLocalization.to_map_eq_zero_iffproof · cited by 2
- Localization.exists_awayMap_injective_of_localRingHom_injectiveproof · cited by 1
- isStronglyTranscendental_mk_of_mem_minimalPrimesproof · cited by 1
- IsLocalization.Away.isIntegral_of_isIntegral_mapproof · cited by 1
- Algebra.FormallySmooth.of_formallySmooth_residueField_tensorproof · cited by 1
- isReduced_localizationPreservesproof · cited by 0
- RingHom.isIntegral_ofLocalizationSpanproof · cited by 0
- AlgebraicGeometry.Scheme.IsGermInjective.Specproof · cited by 0
- Algebra.IsUnramifiedAt.not_minpoly_sq_dvdproof · cited by 0