Theorems · Theorem · commutative algebra
IsPrincipalIdealRing.height_eq_one_of_isMaximal
∀ {R : Type u_1} [inst : CommRing R] [IsDomain R] [IsPrincipalIdealRing R] (m : Ideal R) [m.IsMaximal],
¬IsField R → m.height = 1In a PID that is not a field, every maximal ideal has height one.
- Defined in
- Mathlib.RingTheory.KrullDimension.PID
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites19
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 and proof · cited by 4,985
- Idealstatement and proof · cited by 4,748
- Nat.cast_oneproof · cited by 2,501
- zero_addproof · cited by 2,366
- IsDomainstatement and proof · cited by 2,196
- le_antisymmproof · cited by 2,068
- WithBotproof · cited by 1,498
- WithBot.someproof · cited by 541
- Ideal.IsMaximalstatement and proof · cited by 452
- IsPrincipalIdealRingstatement and proof · cited by 131
- IsFieldstatement and proof · cited by 103
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