Theorems · Theorem · commutative algebra
IsField.localization_map_bijective
∀ {R : Type u_4} {Rₘ : Type u_5} [inst : CommRing R] [inst_1 : CommRing Rₘ] {M : Submonoid R},
0 ∉ M → IsField R → ∀ [inst_2 : Algebra R Rₘ] [IsLocalization M Rₘ], Function.Bijective ⇑(algebraMap R Rₘ)If R is a field, then localizing at a submonoid not containing 0 adds no new elements.
- Defined in
- Mathlib.RingTheory.Localization.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 48 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
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
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Fieldproof · cited by 7,404
- Algebra.algebraMapstatement and proof · cited by 4,706
- mul_oneproof · cited by 3,885
- Submonoidstatement and proof · cited by 3,086
- mul_commproof · cited by 2,262
- mul_assocproof · cited by 1,667
- map_mulproof · cited by 1,137
- nonZeroDivisorsproof · cited by 895
Cited by3
Results whose statement or proof uses this declaration.
- Field.localization_map_bijectiveproof · cited by 0
- IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_botproof · cited by 0
- Polynomial.isMaximal_comap_C_of_isMaximalproof · cited by 0