Theorems · Theorem · commutative algebra
ideal_eq_bot_of_localization
∀ {R : Type u_1} [inst : CommSemiring R] (I : Ideal R),
(∀ (J : Ideal R) (x : J.IsMaximal), IsLocalization.coeSubmodule (Localization.AtPrime J) I = ⊥) → I = ⊥An ideal is trivial if its localization at every maximal ideal is trivial.
- Defined in
- Mathlib.RingTheory.LocalProperties.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
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Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Algebra.algebraMapproof · cited by 4,706
- Eq.leproof · cited by 605
- Ideal.primeComplstatement · cited by 462
- Ideal.IsMaximalstatement and proof · cited by 452
- Localization.AtPrimestatement and proof · cited by 299
- bot_uniqueproof · cited by 57
- IsLocalization.coeSubmodulestatement and proof · cited by 30
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