Theorems · Theorem · commutative algebra
ringKrullDim_eq_one_iff_of_isLocalRing_isDomain
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R] [IsDomain R],
ringKrullDim R = 1 ↔ ¬IsField R ∧ ∀ (x : R), x ≠ 0 → IsLocalRing.maximalIdeal R ≤ (Ideal.span {x}).radical- Cited by
- 0 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsLocalRingIsDomain
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Cites38
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Set.ofPredproof · cited by 6,101
- ENatstatement · cited by 4,985
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- Nat.cast_oneproof · cited by 2,501
- IsDomainstatement and proof · cited by 2,196
- le_antisymmproof · cited by 2,068
- WithBotstatement · cited by 1,498
- le_transproof · cited by 985
- Ideal.spanstatement and proof · cited by 948
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