Theorems · Theorem · commutative algebra
Field.localization_map_bijective
∀ {K : Type u_4} {Kₘ : Type u_5} [inst : Field K] [inst_1 : CommRing Kₘ] {M : Submonoid K},
0 ∉ M → ∀ [inst_2 : Algebra K Kₘ] [IsLocalization M Kₘ], Function.Bijective ⇑(algebraMap K Kₘ)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
- 0 results in Mathlib
- Foundations
- Depth 49 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Algebra.algebraMapstatement · cited by 4,706
- Submonoidstatement and proof · cited by 3,086
- Function.Bijectivestatement · cited by 863
- IsLocalizationstatement and proof · cited by 636
- Field.toIsFieldproof · cited by 18
- IsField.localization_map_bijectiveproof · cited by 3
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