Theorems · Theorem · commutative algebra
eq_zero_of_localization
∀ {R : Type u_1} [inst : CommSemiring R] (r : R),
(∀ (J : Ideal R) (x : J.IsMaximal), (algebraMap R (Localization.AtPrime J)) r = 0) → r = 0- Defined in
- Mathlib.RingTheory.LocalProperties.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Ideal.primeComplstatement and proof · cited by 462
- Ideal.IsMaximalstatement and proof · cited by 452
- Localization.AtPrimestatement and proof · cited by 299
- Algebra.linearMapproof · cited by 157
- OreLocalizationproof · cited by 90
- Module.eq_zero_of_localization_maximalproof · cited by 1
Cited by5
Results whose statement or proof uses this declaration.
- Algebra.FormallyUnramified.isReduced_of_fieldproof · cited by 6
- RingHom.QuasiFinite.ofLocalizationSpanTargetproof · cited by 1
- SymplecticGroup.det_eq_oneproof · cited by 0
- isReduced_ofLocalizationMaximalproof · cited by 0
- ideal_eq_bot_of_localizationproof · cited by 0