Theorems · Theorem · commutative algebra
IsLocalization.mem_map_algebraMap_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] {I : Ideal R} {z : S},
z ∈ Ideal.map (algebraMap R S) I ↔ ∃ x, z * (algebraMap R S) ↑x.2 = (algebraMap R S) ↑x.1- Defined in
- Mathlib.RingTheory.Localization.Ideal
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- SetLike.coeproof · cited by 8,199
- Submoduleproof · cited by 7,192
- Set.imageproof · cited by 5,609
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Submonoidstatement and proof · cited by 3,086
- Submodule.spanproof · cited by 1,504
- Ideal.mapstatement and proof · cited by 692
Cited by9
Results whose statement or proof uses this declaration.
- IsLocalization.mk'_mem_map_algebraMap_iffproof · cited by 5
- Ideal.spanIntNorm_localizationproof · cited by 3
- IsLocalization.under_map_of_isPrimary_disjointproof · cited by 2
- IsLocalization.ideal_eq_iInf_under_map_awayproof · cited by 2
- IsLocalization.AtPrime.under_map_eq_mapproof · cited by 1
- Ideal.iUnion_minimalPrimesproof · cited by 1
- Ideal.exists_mem_span_singleton_map_residueField_eqproof · cited by 0
- IsLocalization.map_infproof · cited by 0