Theorems · Theorem · commutative algebra
IsLocalization.eq_iff_exists
∀ {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] {x y : R}, (algebraMap R S) x = (algebraMap R S) y ↔ ∃ c, ↑c * x = ↑c * yThe kernel of algebraMap is equal to the annihilator by map_units'
- Defined in
- Mathlib.RingTheory.Localization.Defs
- Cited by
- 30 results in Mathlib
- Foundations
- Depth 22 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.eq_iff_existsproof · cited by 9
Cited by30
Results whose statement or proof uses this declaration.
- IsLocalization.isLocalization_of_algEquivproof · cited by 11
- IsLocalization.isLocalization_of_submonoid_leproof · cited by 5
- IsLocalization.mk'_mem_map_algebraMap_iffproof · cited by 5
- IsLocalization.integralClosureproof · cited by 3
- RingHom.surjective_localRingHom_iffproof · cited by 3
- MaximalSpectrum.toPiLocalization_injectiveproof · cited by 3
- IsLocalization.exists_smul_mem_of_mem_adjoinproof · cited by 2
- IsLocalization.isLocalization_of_is_exists_mul_memproof · cited by 2
- IsLocalization.under_map_of_isPrimary_disjointproof · cited by 2
- IsLocalization.ideal_eq_iInf_under_map_awayproof · cited by 2
- Localization.algebraMap_injective_of_span_eq_topproof · cited by 2