Theorems · Theorem · commutative algebra
Ring.DimensionLEOne.localization
∀ {R : Type u_2} (Rₘ : Type u_3) [inst : CommRing R] [IsDomain R] [inst_2 : CommRing Rₘ] [inst_3 : Algebra R Rₘ]
{M : Submonoid R} [IsLocalization M Rₘ], M ≤ nonZeroDivisors R → ∀ [h : Ring.DimensionLEOne R], Ring.DimensionLEOne RₘLocalizing a domain of Krull dimension ≤ 1 gives another ring of Krull dimension ≤ 1.
Note that the same proof can/should be generalized to preserving any Krull dimension,
once we have a suitable definition.
- Defined in
- Mathlib.RingTheory.DedekindDomain.Dvr
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Idealproof · cited by 4,748
- Bot.botproof · cited by 4,720
- Submonoidstatement and proof · cited by 3,086
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsstatement and proof · cited by 895
- Ideal.IsPrimeproof · cited by 827
- IsLocalizationstatement and proof · cited by 636
- Ideal.IsMaximalproof · cited by 452
- Ideal.underproof · cited by 170
- Ideal.IsPrime.ne_topproof · cited by 82
Cited by1
Results whose statement or proof uses this declaration.
- IsLocalization.isDedekindDomainproof · cited by 1