Theorems · Theorem · commutative algebra
Ideal.height_eq_ringKrullDim_iff
∀ {R : Type u_1} [inst : CommRing R] [FiniteRingKrullDim R] [inst_2 : IsLocalRing R] {I : Ideal R} [I.IsPrime],
↑I.height = ringKrullDim R ↔ I = IsLocalRing.maximalIdeal RFor a local ring with finite Krull dimension, a prime ideal has height equal to the Krull dimension if and only if it is the maximal ideal.
- Defined in
- Mathlib.RingTheory.Ideal.Height
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites14
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
- ENatstatement · cited by 4,985
- Idealstatement and proof · cited by 4,748
- WithBotstatement · cited by 1,498
- Ideal.IsPrimestatement and proof · cited by 827
- WithBot.somestatement and proof · cited by 541
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement and proof · cited by 297
- Ideal.heightstatement and proof · cited by 83
- ringKrullDimstatement and proof · cited by 75
- IsLocalRing.eq_maximalIdealproof · cited by 20
- FiniteRingKrullDimstatement and proof · cited by 8
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