Theorems · Theorem · commutative algebra
Ideal.sup_isMaximal_height_eq_ringKrullDim
∀ {R : Type u_1} [inst : CommRing R] [Nontrivial R], ↑(⨆ I, ⨆ (_ : I.IsMaximal), I.height) = ringKrullDim RIn a nontrivial commutative ring R, the supremum of heights of all maximal ideals is
equal to the Krull dimension of R.
- Defined in
- Mathlib.RingTheory.Ideal.Height
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
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
- Top.topproof · cited by 9,680
- ENatstatement and proof · cited by 4,985
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- iSupstatement and proof · cited by 2,415
- le_antisymmproof · cited by 2,068
- le_rflproof · cited by 1,558
- WithBotstatement and proof · cited by 1,498
- eq_or_neproof · cited by 1,117
- WithBot.somestatement and proof · cited by 541
Cited by1
Results whose statement or proof uses this declaration.
- Ring.krullDimLE_of_isLocalization_maximalproof · cited by 1